# Modulational instability in non-Kerr photonic Lieb lattice with
metamaterials

A. K. Shafeeque Ali  Centre for Nonlinear Dynamics, School of Physics,
Bharathidasan University, Tiruchirappalli 620 024, India.  Andrei I. Maimistov
Department of Solid State Physics and Nanostructures, National Nuclear
Research University,
Moscow Engineering Physics Institute, Moscow 115 409.  K. Porsezian
Department of Physics, Pondicherry University, Pondicherry 605 014, India.  A.
Govindarajan  Centre for Nonlinear Dynamics, School of Physics, Bharathidasan
University, Tiruchirappalli 620 024, India.  M. Lakshmanan  Centre for
Nonlinear Dynamics, School of Physics, Bharathidasan University,
Tiruchirappalli 620 024, India.

###### Abstract

We present an analysis of modulational instability of diffractionless waves in
a face-centered square lattice of waveguides featuring non-Kerr nonlinearity,
which are constituted by a combination of positive and negative refractive
indices. The unit cell of the lattice consists of three different waveguides
with different optical properties. The dispersion curve of the lattice
supports flat bands and thereby the base equations describing the model have
particular solutions that correspond to the diffractionless waves propagating
along the waveguides. We also observe a unique ramification of nonlinearities
in controlling the flat bands optically. The diffractionless wave solutions
are derived and the stability of these distributions are investigated in a
nutshell by adopting the standard linear stability approach.

##  I Introduction

In recent studies pertaining to optical lattices, there has been a surge of
interest in the investigation of photonic spectrum with flat band
Flach:Leykam:14 ; Longi:14 ; Maimis:15 ; G:Malomed:16 ; Daniel ; Leykam1 . In
Mukherjee:15 ; Mukherjee:Spracklen:15a  the effect of flat band was
investigated experimentally. As is well-known, the knowledge about the
existence of flat band states in photonic lattices has accelerated quantum
simulation of flat band models in a highly controllable environment
Mukherjee:Spracklen:15a . The appearance of flat band means that for some
particular electromagnetic field distributions inside the waveguide discrete
diffraction is absent. The diffractionless propagation of electromagnetic
waves in waveguide arrays are discussed in Vicencio:14 ; Vicencio:15 ; Fang:15
; Maim:Gabi:16 ; Maim:16 ; Mukherjee2 ; Bastian . The conditions to observe
the propagation of diffractionless modes in the presence of Kerr nonlinearity
in the Lieb lattice have been investigated ppbelicev . Further, non-
Hermiticity-induced flat band in a parity-time symmetric photonic lattice and
controllable localization of light in the non-Hermitian systems as a result of
flat band have been reported Hamidreza . Distortion-free image transmission in
a two-dimensional perovskite-like photonic structure as a result of
superposition of localized flat-band states has been demonstrated Shiqiang .
Also, high-fidelity transmission of the complex patterns in a two-dimensional
pyrochlore-like photonic structure due to the linear superposition of the flat
band eigen modes of the Kagome lattices has been verified Yuanyuan . The
bifurcation of families of localized discrete solitons from the localized
linear modes of the flat band with zero power threshold in a two dimensional
Kagome lattice with defocusing nonlinearity has also been studied vice . The
stability of the flat band solution in a rhombic nonlinear optical waveguide
array breaks down when the intensity per waveguide exceeds the threshold value
maimistov17 .

On the other hand, theoretical investigation on the nonlinear pulse
propagation in waveguide arrays has been receiving considerable attention. For
instance, observation of discrete spatial optical solitons in an array has
been reported Eisenberg . When the initial excitation is not centered on a
waveguide, the discrete solitons in a waveguide array can acquire transverse
momentum and propagate at an angle with respect to the waveguide direction
Morandotti . In the year 2010, the first experimental observation of three-
dimensional light bullets in waveguide arrays featuring quasi-instantaneous
cubic nonlinearity and a periodic, transversally modulated refractive index
has been reported Minardi . The space time coupling in a waveguide array
breaks the spectral symmetry of light bullets to a considerable degree and
modifies their group velocity, leading to superluminal propagation when the
light bullets decay Falk . Also, bright and dark spatial gap solitons have
been demonstrated in waveguide arrays Mandelik . Dispersive shock waves in the
nonlinear waveguide arrays have also been studied experimentally Shu .

Besides the above, study on waveguide arrays with positive and negative
refractive index waveguides have also received considerable attention due to
their unique features. Finite gap solitons observed in the arrays with
positive and negative refractive index waveguides having nonlinearities of
different types show the phenomenon of symmetry breaking in the Fourier space
Zezyulin . Both staggered and unstaggered discrete solitons formed in positive
and negative refractive index material waveguide arrays can become highly
localized states near the zero diffraction points even for low powers
Alexander . The interaction of the nonlinear effects of the channels has great
influence on the generation of the modulation instability Lingling . The
modulation instability of condensate solution for electromagnetic wave
propagating in such waveguide arrays has also been investigated recently shaf
.

Figure 1: Schematic diagram of a two dimensional waveguide array with
alternating sign of refractive index(left) and a unit cell with waveguides A,
B and C of different optical properties(right).

In this paper, we study the propagation of coupled electromagnetic waves in a
two dimensional waveguide array, which consists of waveguides with positive
and negative refractive indices. The cross section of the array is in the form
of face centered square lattice. All waveguides are assumed to be nonlinear
with cubic, quintic and septimal nonlinearities:

$P_{nl}=\chi^{(3)}|E|^{2}-\chi^{(5)}|E|^{4}+\chi^{(7)}|E|^{6},$ (1)

where $\chi^{(n)}$ is the $n$-th order nonlinear susceptibility and $E$ is the
electric field strength of the wave connected to the waveguide.

The structure of two dimensional waveguide array considered in this study can
be realized by arranging unit cells of the array in periodic manner as shown
in Fig. 1. Each unit cell consists three waveguides of different optical
properties. The attainment of modern technologies, such as nanotechnology,
enables the manufacturing of such waveguide arrays with unusual and different
optical properties including negative refraction smith1 ; shelby ; shelby2 .
The phenomenon of negative refraction can be applied in different optical
components for integrated and fiber optics scot ; jason . The nonlinear
properties of metamaterials can be obtained using nonlinear insertions, an
element showing nonlinear response such as diodes to resonant meta atoms
lapine . Nonlinear insertions are suitable to obtain high nonlinear response
with a few watts of power at microwave and lower terahertz frequencies, but
this method fails to develop nonlinear NIM at optical frequencies. At optical
range, desired nonlinear response can be obtained by embedding the metaatoms
into a nonlinear dielectric medium agran . Also, the lossless metamatrials at
optical range can be realized by the implantation of components with active
molecules into the structure of artificial materials Shumin . Moreover,
metamaterial permits engineering of material parameters from their basic
constituents Shad ; Tass . This characteristics provide the tuning of material
parameters at will.

We will theoretically study the existence of flat bands in such a nonlinear
waveguide array and hence focus on the investigation of diffractionless
solution of the system of equations describing the evolution of the
electromagnetic fields. The crucial instability of such diffractionless
solution is also investigated using modulational instability analysis. It is
found that the photonic band structure as well as the stability of flat band
modes are highly dependent on coefficient functions $\kappa(k)$ and higher
order nonlinearities. Hence, the stable propagation of electromagnetic waves
can be achieved by tuning these parameters in the lattice. Also, this study
suggests the possibility for optically controlling the band structures of
waveguide arrays.

The paper is organized as follows. Following a self contained introduction, in
Sec. II the theoretical model and solutions of the problem are presented. The
diffractionless solutions and their stability are studied in Sec. III. In Sec.
IV investigation on modulational instability in metamaterial waveguide arrays
is carried out in detail followed by a short presentation of the summary and
conclusion in Sec. V.

##  II Theoretical Model and Solutions

Consider the waveguide array arranged in a plane as shown in Fig. 1. The cross
section of this configuration has face-centered square lattice structure with
alternating signs of refractive index. The unit cell of the structure consists
of three nonlinear waveguides with cubic, quintic and septimal nonlinearities.
The system of equations describing the evolution of the envelopes of the wave
localized in the waveguide of the unit cell can be derived by using the
procedure developed in shaf  by adopting the tight-binding approximation. The
resulting system of equations is as follows,

$\displaystyle
i(\frac{\partial}{\partial\tau}+\sigma_{1}\frac{\partial}{\partial\zeta})A_{n,m}+\alpha_{1}(B_{n,m}+B_{n-1,m})e^{i\delta_{ba}\zeta}$
$\displaystyle+\alpha_{2}(C_{n,m}+C_{n,m-1})e^{i\delta_{ca}\zeta}+r_{11}|A_{n,m}|^{2}A_{n,m}$
$\displaystyle-r_{12}|A_{n,m}|^{4}A_{n,m}+r_{13}|A_{n,m}|^{6}A_{n,m}=0,$ (2a)
$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma_{2}\frac{\partial}{\partial\zeta}\right)B_{n,m}+\alpha_{1}(A_{n,m}+A_{n+1,m})e^{-i\delta_{ba}\zeta}$
$\displaystyle\qquad\qquad+r_{21}|B_{n,m}|^{2}B_{n,m}-r_{22}|B_{n,m}|^{4}B_{n,m}$
$\displaystyle+r_{23}|B_{n,m}|^{6}B_{n,m}=0,$ (2b) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma_{3}\frac{\partial}{\partial\zeta}\right)C_{n,m}+\alpha_{2}(A_{n,m}+A_{n,m+1})e^{-i\delta_{ca}\zeta}$
$\displaystyle+r_{31}|C_{n,m}|^{2}C_{n,m}-r_{32}|C_{n,m}|^{4}C_{n,m}$
$\displaystyle+r_{33}|C_{n,m}|^{6}C_{n,m}=0.$ (2c)

Here $\sigma_{j}$ represents the sign of refractive index of individual
waveguides. Hence, for a positive refractive index waveguide $\sigma_{j}=1$,
and for a negative refractive index waveguide $\sigma_{j}=-1$. The pair
$(n,m)$ of integers stands for the unit cell label in the two dimensional
lattice. Also $A_{n,m}$, $B_{n,m}$ and $C_{n,m}$ are the normalized envelopes
of the associated wave localized in the appropriate waveguide of the unit cell
with indices $(n,m)$. Further, $\delta_{ba}=\beta_{b}-\beta_{a}$ and
$\delta_{ca}=\beta_{c}-\beta_{a}$ are the mismatch between the wave numbers
(propagation constants) $\beta_{a}$, $\beta_{b}$ and $\beta_{c}$. The
parameters $\alpha_{1}$ and $\alpha_{2}$ indicate the coupling strengths
between neighboring waveguides. Also, $r_{11}$, $r_{21}$ and $r_{31}$ are
cubic nonlinear coefficients, $r_{12}$, $r_{22}$ and $r_{32}$ are quintic
nonlinear coefficients and $r_{13}$, $r_{23}$ and $r_{33}$ are septimal
nonlinear coefficients.

Now, let us choose for convenience the various nonlinear coefficients as
$r_{11}=r_{21}=r_{31}=R_{1}$, $r_{12}=r_{22}=r_{32}=R_{2}$ and
$r_{13}=r_{23}=r_{33}=R_{3}$. Also, we introduce the following transformations
of the fields,

$\displaystyle A_{n,m}=\tilde{A}_{n,m}e^{i\delta_{ba}\zeta/2},\quad
B_{n,m}=\tilde{B}_{n,m}e^{-i\delta_{ba}\zeta/2},$ $\displaystyle\qquad\quad
C_{n,m}=\tilde{C}_{n,m}e^{i(\delta_{ba}/2-\delta_{ca})\zeta},$

so that the system of Eqs. (2) takes the following form,

$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma_{1}\frac{\partial}{\partial\zeta}\right)\tilde{A}_{n,m}-\frac{1}{2}\sigma_{1}\delta_{ba}\tilde{A}_{n,m}$
$\displaystyle+\alpha_{1}(\tilde{B}_{n,m}+\tilde{B}_{n-1,m})+\alpha_{2}(\tilde{C}_{n,m}+\tilde{C}_{n,m-1})$
$\displaystyle+R_{1}|\tilde{A}_{n,m}|^{2}\tilde{A}_{n,m}-R_{2}|\tilde{A}_{n,m}|^{4}\tilde{A}_{n,m}$
$\displaystyle+R_{3}|\tilde{A}_{n,m}|^{6}\tilde{A}_{n,m}=0,$ (3a)
$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma_{2}\frac{\partial}{\partial\zeta}\right)\tilde{B}_{n,m}+\frac{1}{2}\sigma_{2}\delta_{ba}\tilde{B}_{n,m}$
$\displaystyle+\alpha_{1}(\tilde{A}_{n,m}+\tilde{A}_{n+1,m})+R_{1}|\tilde{B}_{n,m}|^{2}\tilde{B}_{n,m}$
$\displaystyle-
R_{2}|\tilde{B}_{n,m}|^{4}\tilde{B}_{n,m}+R_{3}|\tilde{B}_{n,m}|^{6}\tilde{B}_{n,m}=0,$
(3b) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma_{3}\frac{\partial}{\partial\zeta}\right)\tilde{C}_{n,m}-\sigma_{3}\varphi_{0}\tilde{C}_{n,m}+$
$\displaystyle\qquad\qquad+\alpha_{2}(\tilde{A}_{n,m}+\tilde{A}_{n,m+1})+R_{1}|\tilde{C}_{n,m}|^{2}\tilde{C}_{n,m}$
$\displaystyle-
R_{2}|\tilde{C}_{n,m}|^{4}\tilde{C}_{n,m}+R_{3}|\tilde{C}_{n,m}|^{6}\tilde{C}_{n,m}=0,$
(3c)

where $\varphi_{0}=\delta_{ba}/2-\delta_{ca}$. Let us assume the waveguides B
and C are identical, admitting the same propagation constants
($\beta_{b}=\beta_{c}$), so that $\delta_{ba}/2=\delta_{ca}/2=\Delta$, where
$\Delta=-\varphi_{0}$. Also, let us consider the case of
$\sigma_{2}=\sigma_{3}\equiv\sigma=\pm 1$ and $\sigma_{1}=1$. Thus the new
system of equations reads as

$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\frac{\partial}{\partial\zeta}\right)\tilde{A}_{n,m}-\Delta\tilde{A}_{n,m}$
$\displaystyle+\alpha_{1}(\tilde{B}_{n,m}+\tilde{B}_{n-1,m})$
$\displaystyle+\alpha_{2}(\tilde{C}_{n,m}+\tilde{C}_{n,m-1})+R_{1}|\tilde{A}_{n,m}|^{2}\tilde{A}_{n,m}$
$\displaystyle-
R_{2}|\tilde{A}_{n,m}|^{4}\tilde{A}_{n,m}+R_{3}|\tilde{A}_{n,m}|^{6}\tilde{A}_{n,m}=0,$
(4a) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{B}_{n,m}+\sigma\Delta\tilde{B}_{n,m}+$
$\displaystyle\qquad\qquad+\alpha_{1}(\tilde{A}_{n,m}+\tilde{A}_{n+1,m})+R_{1}|\tilde{B}_{n,m}|^{2}\tilde{B}_{n,m}$
$\displaystyle-
R_{2}|\tilde{B}_{n,m}|^{4}\tilde{B}_{n,m}+R_{3}|\tilde{B}_{n,m}|^{6}\tilde{B}_{n,m}=0,$
(4b) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{C}_{n,m}+\sigma\Delta\tilde{C}_{n,m}+$
$\displaystyle+\alpha_{2}(\tilde{A}_{n,m}+\tilde{A}_{n,m+1})+R_{1}|\tilde{C}_{n,m}|^{2}\tilde{C}_{n,m}$
$\displaystyle-
R_{2}|\tilde{C}_{n,m}|^{4}\tilde{C}_{n,m}+R_{3}|\tilde{C}_{n,m}|^{6}\tilde{C}_{n,m}=0.$
(4c)

The next approximation is the zero mismatch ($\Delta=0$), which indicates that
all the three waveguides of the unit cell posses the same propagation
constant. Then the system of equations takes the following forms,

$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\frac{\partial}{\partial\zeta}\right)\tilde{A}_{n,m}+\alpha_{1}(\tilde{B}_{n,m}+\tilde{B}_{n-1,m})$
$\displaystyle+\alpha_{2}(\tilde{C}_{n,m}+\tilde{C}_{n,m-1})+R_{1}|\tilde{A}_{n,m}|^{2}\tilde{A}_{n,m}$
$\displaystyle-
R_{2}|\tilde{A}_{n,m}|^{4}\tilde{A}_{n,m}+R_{3}|\tilde{A}_{n,m}|^{6}\tilde{A}_{n,m}=0,$
(5a) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{B}_{n,m}$
$\displaystyle+\alpha_{1}(\tilde{A}_{n,m}+\tilde{A}_{n+1,m})+R_{1}|\tilde{B}_{n,m}|^{2}\tilde{B}_{n,m}$
$\displaystyle-
R_{2}|\tilde{B}_{n,m}|^{4}\tilde{B}_{n,m}+R_{3}|\tilde{B}_{n,m}|^{6}\tilde{B}_{n,m}=0,$
(5b) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{C}_{n,m}$
$\displaystyle+\alpha_{2}(\tilde{A}_{n,m}+\tilde{A}_{n,m+1})+R_{1}|\tilde{C}_{n,m}|^{2}\tilde{C}_{n,m}$
$\displaystyle-
R_{2}|\tilde{C}_{n,m}|^{4}\tilde{C}_{n,m}+R_{3}|\tilde{C}_{n,m}|^{6}\tilde{C}_{n,m}=0.$
(5c)

Figure 2: (Color online.) The linear dispersion relation of Lieb lattice
featuring two-dimensional waveguide arrays. (a) shows the three bands in the
first Brillouin zone ($k_{1,2}\in[-\pi,\pi]$) and (b) depicts the same in the
second Brillouin regime ($k_{1,2}\in[-2\pi,2\pi]$). Here the flat band (drawn
in green color) is separated by two dispersive (upper and lower) bands,
indicated by the color combinations of yellow, red and blue, purple
respectively. (c) portrays a band structure of $n=20$ unit cells with
$k_{1,2}\in[0,2\pi]$. The system parameters are $\alpha_{1}=\alpha_{2}=l=1$,
and $R_{1}=R_{2}=R_{3}\equiv R=0$.

Let us now discuss the dispersion relations of the waves which are governed by
Eqs. (5). Consider the quasi-harmonic waves

$\displaystyle\tilde{A}_{mn}=A_{0}e^{-i\omega\tau+ik_{z}\zeta+ik_{1}n+ik_{2}m},$
(6a)
$\displaystyle\tilde{B}_{mn}=B_{0}e^{-i\omega\tau+ik_{z}\zeta+ik_{1}n+ik_{2}m},$
(6b)
$\displaystyle\tilde{C}_{mn}=C_{0}e^{-i\omega\tau+ik_{z}\zeta+ik_{1}n+ik_{2}m},$
(6c)

where $k_{z}$ is a small correction to the constant of propagation along the
waveguide, $k_{1}=k_{x}l$ and $k_{2}=k_{y}l$ are normalized wave-numbers and
$l$ is the lattice parameter. The quantities $k_{x}$ and $k_{y}$ are the quasi
(Bloch) momenta of the 2D Lieb lattice. Substitution of these expressions in
Eq. (5) results in the following system of algebraic equations,

$\displaystyle(\omega-
k_{z}+f_{1})A_{0}+\kappa_{1}^{*}B_{0}+\kappa_{2}^{*}C_{0}=0,$ (7a)
$\displaystyle\kappa_{1}A_{0}+(\omega-\sigma k_{z}+f_{2})B_{0}=0,$ (7b)
$\displaystyle\kappa_{2}A_{0}+(\omega-\sigma k_{z}+f_{3})C_{0}=0,$ (7c)

where the nonlinearity contributions in the dispersion relations are

$\displaystyle f_{1}=R_{1}|A_{0}|^{2}-R_{2}|A_{0}|^{4}+R_{3}|A_{0}|^{6},$ (8a)
$\displaystyle f_{2}=R_{1}|B_{0}|^{2}-R_{2}|B_{0}|^{4}+R_{3}|B_{0}|^{6},$ (8b)
$\displaystyle f_{3}=R_{1}|C_{0}|^{2}-R_{2}|C_{0}|^{4}+R_{3}|C_{0}|^{6}.$ (8c)

In Eq. (7) we have also introduced of the normalized wave-numbers,

$\displaystyle\kappa_{1}=\alpha_{1}(1+e^{ik_{1}}),$ (9a)
$\displaystyle\kappa_{2}=\alpha_{2}(1+e^{ik_{2}}).$ (9b)

Non-zero solutions of Eq. (7) exist if the determinant associated with this
system of equations is equal to zero. It results in the dispersion equation,

$\displaystyle(\omega-k_{z}+f_{1})(\omega-\sigma k_{z}+f_{2})(\omega-\sigma
k_{z}+f_{3})$ $\displaystyle-(\omega-\sigma
k_{z}+f_{2})|\kappa_{2}|^{2}-(\omega-\sigma k_{z}+f_{3})|\kappa_{1}|^{2}=0.$
(10)

The general solution to this equation is given in Appendix 1. For a particular
case $f_{2}=f_{3}$, the dispersion relation
$\omega=\omega(\kappa_{1},\kappa_{2},k_{z};A_{0},B_{0},C_{0})$ becomes a
factorized one, which results in the following relations,

$\displaystyle\omega=\sigma k_{z}-f_{2},$ (11a) $\displaystyle(\omega-
k_{z}+f_{1})(\omega-\sigma
k_{z}+f_{2})-(|\kappa_{1}|^{2}+|\kappa_{2}|^{2})=0.$ (11b)

The first branch of the dispersion relation, i.e. Eq. (11a), corresponds to
diffractionless wave propagation (as $d\omega/dk$ is independent of the
wavenumber). At $\sigma=1$ it is a forward wave, and at $\sigma=-1$ it is a
backward one. Note that the backward propagation of the wave is due to the
negative refraction in the metamaterial.

Figure 3: (Color online.) The nonlinear dispersion relation of Lieb lattice
with 20 unit cells ($n=20$) for the same parameters as in Fig. 2 except, (a)
$R_{2}=R_{3}=0,R_{1}=1$, (b) $R_{1}=R_{2}=1,R_{3}=0$, and (c)
$R_{1}=R_{2}=1=R_{3}=1$ with $|A_{0}|^{2}=|B_{0}|^{2}=|C_{0}|^{2}=1$. Bottom
panels indicate the corresponding three dimensional structure for a finite
Lieb lattice.

We first present the dispersion characteristics of Eq. (11) for the linear
system. Then for a linear system $R_{1}=R_{2}=R_{3}\equiv R=0$ and the case is
depicted in Fig. 2. It is to be noted that to sketch the dispersion relations,
we have considered the frequency of the Lieb lattice as a function of the two
dimensional Bloch wave vectors $k_{1}$ and $k_{2}$ by neglecting the small
correction in the propagation constant ($k_{z}=0$). One can clearly notice
from Fig. 2 drawn in the first Brillouin zone ($k_{1,2}\in[-\pi,\pi]$) that
the system supports the typical dispersion curves with three energy bands
including a perfectly flat band, which is identical to the energy bands
observed in photonic Lieb lattices Mukherjee:Spracklen:15a ; ppbelicev  and
Kagome lattices in addition to the two dispersive bands liang . Such a
characteristics where all the three bands get into contact with the symmetric
band, that is the flat band is known as the particle-hole symmetry analogous
to the quantum version. Also, this flat band is a manifestation of degenerate
state meaning that it is static and will not contribute to any transport of
localized state. This implies that the localized states of flat bands are
diffractionless since their group-velocity is zero. When the dispersion
relation is plotted in the second Brillouin zone ($k_{1,2}\in[-2\pi,2\pi]$),
quite a number of unique features of bands gets revealed. For instance, one
can observe Dirac cones (marked with circles in Fig. 2(a)) of the conical
dispersive bands intersect with the flat band at the corner of the first
Brillouin zone. Also, Tamm-like edge states of dispersive bands result in a
van Hove singularity in the given three bands (see blue colored arrow marks in
Fig. 2(a)). Further, the band structure shown in Fig. 2(c) (obtained for 20
unit cells ($n=20$)) clearly reveals the topologically protected solid edge
states (drawn in green color curve) in addition to the typical bulk states
(black lined curves).

The nonlinear dispersion relation is shown in Fig. 3. As the cubic
nonlinearity acts alone, the flat band shifts towards low (negative) frequency
from zero one as depicted in Figs. 3(a) and 3(d) while both the dispersion
(conical) bands get shifted towards the negative frequency from the positive
side. On the other hand, if the quintic nonlinearity is invoked along with the
cubic one, one can observe that the shift of flat band is quite opposite to
the cubic one. When the system includes the septimal nonlinearity, besides the
cubic and quintic ones, the flat band and dispersive bands completely get
drifted towards the negative frequency. Hence it is clear that when we include
higher order nonlinearities such as cubic, quintic and septimal ones, the
degeneracy of the flat band is even reduced and located on the top of conical
bands. Thus, one can conclude that the inclusion of nonlinearity shifts the
flat band from zero frequency to higher as well lower values depending upon
the type of nonlinearities. In other words, the high intensity optical light
alters the location of degenerate and dispersive bands. These ramifications
clearly suggest the possibility of optically controlling the band structure of
a Lieb waveguide array.

In the linear regime with the following approximations, Eqs. (5) have
particular solutions, which are written as

$\displaystyle(\mathrm{A})~{}~{}\tilde{A}_{n,m}=0,\quad\alpha_{1}\tilde{B}_{n,m}=-\alpha_{2}\tilde{C}_{n,m-1},$
$\displaystyle\qquad\qquad\alpha_{1}\tilde{B}_{n-1,m}=-\alpha_{2}\tilde{C}_{n,m},$
(12)
$\displaystyle(\mathrm{B})~{}~{}~{}\tilde{A}_{n,m}=0,\quad\alpha_{1}\tilde{B}_{n,m}=-\alpha_{1}\tilde{B}_{n-1,m},$
$\displaystyle\qquad\qquad\alpha_{2}\tilde{C}_{n,m}=-\alpha_{2}\tilde{C}_{n,m-1},$
(13)
$\displaystyle(\mathrm{C})\tilde{A}_{n,m}=(-1)^{n+m}\tilde{A},\quad\tilde{B}_{n,m}=(-1)^{n+m}\tilde{B},$
$\displaystyle\qquad\qquad\tilde{C}_{n,m}=(-1)^{n+m}\tilde{C}.$ (14)

It is worthwhile to mention that the diffractionless propagation of
electromagnetic waves in the two dimensional waveguide arrays under
consideration is described by these solutions in such a linear approximation.
Similar behavior in the nonlinear case will be considered in the next section.

##  III The diffractionless solutions and their stability

(a) $R_{1}=1$ and $R_{2}=R_{3}=0$

(b) $R_{1}=R_{2}=R_{3}=1$

(c) $R_{1}=-R_{2}=1$ and $R_{3}=0$

(d) $R_{1}=-R_{2}=R_{3}=1$

(e) $R_{1}=R_{2}=-R_{3}=1$

Figure 4: (Color online.) The MI gain spectra versus $K$ as a function of
$p=q=S$ for different combinations of nonlinear coefficients.

To find the nonlinear analogy of Eqs. (12)–(14), let us suppose that
$\tilde{A}_{n,m}$ is zero at all $\zeta$ and $\tau$. Then the system of Eqs.
(5) is reduced to following equations,

$\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{B}_{n,m}+R_{1}|\tilde{B}_{n,m}|^{2}\tilde{B}_{n,m}$
$\displaystyle-
R_{2}|\tilde{B}_{n,m}|^{4}\tilde{B}_{n,m}+R_{3}|\tilde{B}_{n,m}|^{6}\tilde{B}_{n,m}=0,$
(15a) $\displaystyle
i\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right)\tilde{C}_{n,m}+R_{1}|\tilde{C}_{n,m}|^{2}\tilde{C}_{n,m}$
$\displaystyle-
R_{2}|\tilde{C}_{n,m}|^{4}\tilde{C}_{n,m}+R_{3}|\tilde{C}_{n,m}|^{6}\tilde{C}_{n,m}=0.$
(15b)

The homogeneous solution of these equations have the following form

$\displaystyle\tilde{A}_{n,m}=0,\quad\tilde{B}_{n,m}=B_{0}e^{i(R_{1}|B_{0}|^{2}-R_{2}|B_{0}|^{4}+R_{3}|B_{0}|^{6})\zeta},$
$\displaystyle\tilde{C}_{n,m}=C_{0}e^{i(R_{1}|B_{0}|^{2}-R_{2}|B_{0}|^{4}+R_{3}|B_{0}|^{6})\zeta},$
(16)

where the constraints $\alpha_{2}=\alpha_{1}=1$ and $|C_{0}|=|B_{0}|$ are
used.

To study the stability of the associated solutions (III), the appropriately
perturbed amplitudes must be introduced. For instance we choose,

$\displaystyle\tilde{A}_{n,m}=a_{n,m}e^{i\varphi},\quad\tilde{B}_{n,m}=(B_{0}+b_{n,m})e^{i\varphi},$
$\displaystyle\tilde{C}_{n,m}=(-B_{0}+c_{n,m})e^{i\varphi},$

where
$\partial\varphi/\partial\xi=R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$ .
The linearization of the system of Eqs. (5) results in the following equations

$\displaystyle i\frac{\partial
a_{nm}}{\partial\eta}+a_{nm}(R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6})$
$\displaystyle+(b_{n,m}+b_{n-1,m})+(c_{n,m}+c_{n,m-1})=0,$ (17a)
$\displaystyle i\frac{\partial
b_{nm}}{\partial\xi}+(a_{nm}+a_{n+1,m})+R_{1}B_{0}^{2}\left(b_{nm}^{*}+b_{nm}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(b_{nm}^{*}+b_{nm}\right)+R_{3}B_{0}^{6}\left(b_{nm}^{*}+b_{nm}\right)=0,$
(17b) $\displaystyle i\frac{\partial
c_{nm}}{\partial\xi}+(a_{nm}+a_{n,m+1})+R_{1}B_{0}^{2}\left(c_{nm}^{*}+c_{nm}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(c_{nm}^{*}+c_{nm}\right)+R_{3}B_{0}^{6}\left(c_{nm}^{*}+c_{nm}\right)=0,$
(17c)

where

$\frac{\partial}{\partial\eta}=\left(\frac{\partial}{\partial\tau}+\frac{\partial}{\partial\zeta}\right),\quad\frac{\partial}{\partial\xi}=\left(\frac{\partial}{\partial\tau}+\sigma\frac{\partial}{\partial\zeta}\right).$

Disinflation of the system of linear equations can be done by using the
Fourier transformations,

$\displaystyle a_{nm}=\sum_{p,q}\left(a_{pq}e^{ipn+iqm}+\tilde{a}_{pq}e^{-ipn-
iqm}\right),$ $\displaystyle
b_{nm}=\sum_{p,q}\left(b_{pq}e^{ipn+iqm}+\tilde{b}_{pq}e^{-ipn-iqm}\right),$
$\displaystyle c_{nm}=\sum_{p,q}\left(c_{pq}e^{ipn+iqm}+\tilde{c}_{pq}e^{-ipn-
iqm}\right).$

Equations for the Fourier amplitudes $a_{pq}$, $b_{pq}$, $c_{pq}$,
$\bar{a}_{pq}$, $\bar{b}_{pq}$ and $\bar{c}_{pq}$ follow from Eqs. (17) as,

$\displaystyle i\frac{\partial
a_{pq}}{\partial\eta}+a_{pq}(R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6})+\kappa_{1}b_{pq}+\kappa_{2}c_{pq}=0,$
(18a) $\displaystyle i\frac{\partial
b_{pq}}{\partial\xi}+\kappa_{1}^{*}a_{pq}+R_{1}B_{0}^{2}\left(b_{pq}+\bar{b}_{pq}^{*}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(b_{pq}+\bar{b}_{pq}^{*}\right)+R_{3}B_{0}^{6}\left(b_{pq}+\bar{b}_{pq}^{*}\right)=0,$
(18b) $\displaystyle i\frac{\partial
c_{pq}}{\partial\xi}+\kappa_{2}^{*}a_{pq}+R_{1}B_{0}^{2}\left(c_{pq}+\bar{c}_{pq}^{*}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(c_{pq}+\bar{c}_{pq}^{*}\right)+R_{3}B_{0}^{6}\left(c_{pq}+\bar{c}_{pq}^{*}\right)=0,$
(18c) $\displaystyle
i\frac{\partial\bar{a}_{pq}}{\partial\eta}+\bar{a}_{pq}(R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6})+\kappa_{1}^{*}\bar{b}_{pq}+\kappa_{2}^{*}\bar{c}_{pq}=0,$
(18d) $\displaystyle
i\frac{\partial\bar{b}_{pq}}{\partial\xi}+\kappa_{1}\bar{a}_{pq}+R_{1}B_{0}^{2}\left(b_{pq}^{*}+\bar{b}_{pq}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(b_{pq}^{*}+\bar{b}_{pq}\right)+R_{3}B_{0}^{6}\left(b_{pq}^{*}+\bar{b}_{pq}\right)=0,$
(18e) $\displaystyle
i\frac{\partial\bar{c}_{pq}}{\partial\xi}+\kappa_{2}\bar{a}_{pq}+R_{1}B_{0}^{2}\left(c_{pq}^{*}+\bar{c}_{pq}\right)$
$\displaystyle-
R_{2}B_{0}^{4}\left(c_{pq}^{*}+\bar{c}_{pq}\right)+R_{3}B_{0}^{6}\left(c_{pq}^{*}+\bar{c}_{pq}\right)=0.$
(18f)

Here, the coefficient functions are denoted as

$\displaystyle\qquad\kappa_{1}=\kappa(p),\quad\kappa_{2}=\kappa(q),$
$\displaystyle\kappa(k)=1+e^{-ik}=2\cos(k/2)e^{-ik/2}.$

In order to solve the above system of six linear differential equations, we
assume the following plane wave ansatz,

$\displaystyle
a_{pq}=ae^{iK\zeta-i\Omega\tau},\quad\bar{a}_{pq}=\bar{a}e^{-iK\zeta+i\Omega\tau},$
$\displaystyle
b_{pq}=be^{iK\zeta-i\Omega\tau},\quad\bar{b}_{pq}=\bar{b}e^{-iK\zeta+i\Omega\tau},$
$\displaystyle
c_{pq}=ce^{iK\zeta-i\Omega\tau},\quad\bar{c}_{pq}=\bar{c}e^{-iK\zeta+i\Omega\tau}.$

Substituting these expressions in Eq. (18), we obtain a set of linearly
coupled algebraic equations for $a$, $\bar{a}$, b, $\bar{b}$, c and $\bar{c}$.
This set has nontrivial solutions only when the 6x6 determinant formed by the
coefficient matrix vanishes as given below:

$\left(\begin{array}[]{cccccc}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}&\epsilon_{14}&\epsilon_{15}&\epsilon_{16}\\\
\epsilon_{21}&\epsilon_{22}&\epsilon_{23}&\epsilon_{24}&\epsilon_{25}&\epsilon_{26}\\\
\epsilon_{31}&\epsilon_{32}&\epsilon_{33}&\epsilon_{34}&\epsilon_{35}&\epsilon_{36}\\\
\epsilon_{41}&\epsilon_{42}&\epsilon_{43}&\epsilon_{44}&\epsilon_{45}&\epsilon_{46}\\\
\epsilon_{51}&\epsilon_{52}&\epsilon_{53}&\epsilon_{54}&\epsilon_{55}&\epsilon_{56}\\\
\epsilon_{61}&\epsilon_{62}&\epsilon_{63}&\epsilon_{64}&\epsilon_{65}&\epsilon_{66}\\\
\end{array}\right)\left(\begin{array}[]{c}a\\\ b\\\ c\\\ \bar{a}\\\ \bar{b}\\\
\bar{c}\\\ \end{array}\right)=0,$

where $\epsilon_{11}=-K+\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{12}=\kappa_{1}$, $\epsilon_{13}=\kappa_{2}$, $\epsilon_{14}=0$,
$\epsilon_{15}=0$ $\epsilon_{16}=0$, $\epsilon_{21}=\kappa_{1}^{*}$,
$\epsilon_{22}=-\sigma K+\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{23}=0$, $\epsilon_{24}=0$,
$\epsilon_{25}=R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{26}=0$, $\epsilon_{31}=\kappa_{2}^{*}$, $\epsilon_{32}=0$,
$\epsilon_{33}=-\sigma K+\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{34}=0$, $\epsilon_{35}=0$,
$\epsilon_{36}=R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{41}=0$, $\epsilon_{42}=0$, $\epsilon_{43}=0$,
$\epsilon_{44}=K-\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{45}=\kappa_{1}^{*}$, $\epsilon_{46}=\kappa_{2}^{*}$,
$\epsilon_{51}=0$,
$\epsilon_{52}=R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{53}=0$, $\epsilon_{54}=\kappa_{1}$, $\epsilon_{55}=\sigma
K-\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$, $\epsilon_{56}=0$,
$\epsilon_{61}=0$, $\epsilon_{62}=0$,
$\epsilon_{63}=R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$,
$\epsilon_{64}=\kappa_{2}$, $\epsilon_{65}=0$, $\epsilon_{66}=\sigma
K-\Omega+R_{1}B_{0}^{2}-R_{2}B_{0}^{4}+R_{3}B_{0}^{6}$.

The determinant of the system of algebraic equations must be equal to zero
which results in the dispersion relation
$\Omega=\Omega(K,q,p;R_{1},R_{2},R_{3},B_{0})$ through which one can measure
the instability gain spectra as $G({\Omega})=|\Im{\Omega_{max}}|$. Where,
$\Im{\Omega_{max}}$ denotes the imaginary part of $\Omega_{max}$, where
$\Omega_{max}$ is the root of the polynomial with largest value.

##  IV Modulational instability in metamaterial waveguide arrays

(a) $R_{1}=1$ and $R_{2}=R_{3}=0$

(b) $R_{1}=-R_{2}=1$ and $R_{3}=0$

(c) $R_{1}=-R_{2}=R_{3}=1$

(d) $R_{1}=R_{2}=R_{3}=1$

Figure 5: (Color online.) Periodic MI gain spectra in the $p-q$ plane.

In this section we discuss the modulational instability of the flat band modes
in the waveguide arrays with negative index material channels in detail. Let
us choose initial power of incident wave, $P=B_{0}^{2}=1$. It is well known
that higher order nonlinearities can considerably influence the system
dynamics triki2016 ; triki2017 ; Raja1  and the modulation instability gain
spectra in any system. Negative index materials embedded in cubic and quintic
nonlinear media give more ways to manipulate and control modulation
instability and hence the soliton formation sharma1 . Quintic nonlinearity
plays a major role in the formation of gap solitons in fiber Bragg grating
kp111 . The symmetric and asymmetric modulational instability growth rates
have been observed in a zigzag array of nonlinear waveguides with the
alternating signs of refractive indices ADD . The self-focussing and self-
defocussing nonlinearity0 of positive and negative refractive index waveguides
affect the modulational instability gain of the array ADD1 . Modulational
instability in the presence of higher order nonlinearities may be beneficial
to the generation of high repetition rate pulse trains in oppositely directed
couplers malo . In the same way, we investigate the influence of quintic and
septimal nonlinearities on modulational instability in metamaterial waveguide
arrays.

It is well known that modulational instability is a precursor for the
formation of solitons. The modulational instability of diffractionless modes
can lead to bifurcations of the modes to soliton-like solutions. Here we have
given a special emphasis to analyze the influence of higher-nonlinear effects
on modulational instability of diffractionless modes agp1 ; agp2 . Fig. 4
depicts the instability gain spectra of diffractionless solution versus
perturbation wave vector $K$ as a function of $p=q=S$ for different possible
combinations of cubic, quintic and septimal nonlinearities. Fig. 4(a), which
depicts the Kerr nonlinear case, it is clear from the figure that the MI
spectra is periodic in $p=q=S$ with a period $2\pi$ (in the second Brillouin
zone). In the Kerr nonlinear case each period consists of two instability
regions, which are separated by a stable region located at $p=q=S=n\pi$ with
$n=0,1,2,3...$.

Now let us consider the influence of higher order nonlinearities originating
from fifth and seventh order ( $\chi^{(5)}$ and $\chi^{(7)}$)
susceptibilities. Fig. 4(b) represents the MI gain spectra with cubic, quintic
and septimal nonlinearities. The quintic nonlinearity is of defocusing type
whereas the cubic and septimal nonlinearities are of focusing types. Comparing
with the cubic case (Fig. 4(a)), here one can see the enhancement of MI gain
spectrum by increasing the gain and band width. In this case too, the periodic
MI vanishes when the parameter $S$ satisfies the condition $S=n\pi$ with
$n=0,1,2,3...$. Hence, the presence of non-Kerr nonlinearity enhances the
modulational instability of diffractionless waves and provides more ways to
manipulate solitons.

Fig. 4(c) depicts the case with focusing cubic and quintic nonlinearities
alone. In this case also MI is periodic in $S$ with a period $2\pi$. It is
interesting to note that compared to the previous cases (Figs. 4(a) and 4(b))
here the MI gain is present for $S$ values of even integral multiples of
$\pi$. The MI gain vanishes when $p=q=S=n\pi$ with $n=1,3,5...$. The presence
of focusing septimal nonlinearity enhances the MI by increasing the gain and
enlarging the instability band as portrayed in Fig. 4(d). The MI gain spectra
for the case of focusing cubic and defocusing quintic and septimal
nonlinearities is depicted in Fig. 4(e). Here also one can understand the role
of higher order nonlinearity in the enhancement of MI. Therefore, the stable
propagation diffractionless mode and formation of localized soliton-like
structure in non-Kerr photonic Lieb lattice with metamaterials can be achieved
by properly tuning higher-order nonlinearities.

Fig. 5 depicts the MI gain spectra in the $p-q$ plane for different
combinations of nonlinearities. It is clear from Fig. 5 that, the gain spectra
are periodic in the $p-q$ plane with period $2\pi$. Stable propagation of
diffractionless wave is observed when both $p$ and $q$ simultaneously satisfy
the condition $p=q=n\pi$, $n=0,1,2,3...$ in Figs. 5(a) and 5(d), where Fig.
5(a) corresponds to the cubic nonlinear case and in Fig. 5(d) cubic as well as
septimal nonlinearities are of focusing types whereas quintic nonlinearity is
of defocusing type. But the values of $n$ are odd numbers in Figs. 5(b) and
5(c). It is either due to the presence of focusing quintic nonlinearity (Fig.
5(b)) or by the defocusing septimal nonlinearity Figs. 5(c).

One can thus conclude that the stability of diffractionless waves propagating
in a waveguide array with alternating signs of refractive index is highly
influenced by the values of higher order nonlinear coefficients and the
normalized coefficient functions $\kappa(k)$. Stable propagation of
electromagnetic wave can be achieved by controlling these parameters in the
lattice. Also, as the modulation instability is a precursor for the pattern
formations in the form ultra-short pulses, controlling of modulation
instability with these parameters provides better ways to generate and
manipulate solitons in arrays of waveguides with negative index of refraction.

##  V Conclusion

In summary, we have investigated the propagation of flat band modes in a face-
centered square lattice of waveguide array, which is featured by positive and
negative refractive indices. We have considered three different waveguides
with different optical properties in a unit cell of the lattice. The study
shows that the lattice supports dispersion curve with flat band and hence, it
can bear diffractionless wave propagation. The photonic band structure as well
as the stability of flat band modes are highly dependent on coefficient
functions $\kappa(k)$ and higher order nonlinearities. Hence, the stable
propagation of electromagnetic waves can be achieved by tuning these
parameters in the lattice. Also, this study suggests the possibility for
optically controlling the band structures of waveguide arrays. We thus
anticipate that the present investigation can pave a new roadmap on the
nonlinear flat band modes which can be a promising candidate to control light
by light in arrays of combined positive-negative index waveguides.

##  Acknowledgement

A. K. S. is very much grateful to Dr. K. Porsezian, who although is no longer
with us but continues to inspire with his example and dedication, and who
ignited the theme of the present work. The work of A.K.S. is supported by the
University Grants Commission (UGC), Government of India, through a D. S.
Kothari Post Doctoral Fellowship in Sciences. The research of A.I.M. was
supported by the Russian Foundation for Basic Research (Grant No.
18-02-00278). A. G. is supported by the University Grants Commission (UGC),
Government of India, through a D. S. Kothari Post Doctoral Fellowship in
Sciences. M. L. is supported by a DST-SERB Distinguished Fellowship (Grant No.
SB/DF/04/2017).

## Appendix

In this Appendix, we provide the general solution to the dispersion equation
(II). It reads as

$\omega_{1}=\frac{1}{3}N_{1}-[2^{1/3}(N_{2}+3N_{3})]/\\{3[N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]^{1/3}\\}$
$+\frac{1}{2^{1/3}3}\\{N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]\\}^{1/3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A1)$

$\omega_{2}=\frac{1}{3}N_{1}+[(1+i\sqrt{3})(N_{2}+3N_{3})]/\\{2^{2/3}3[N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]^{1/3}\\}$
$-\frac{1}{62^{1/3}}(1-i\sqrt{3})\\{N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]\\}^{1/3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A2)$

$\omega_{3}=\frac{1}{3}N_{1}-[(1-i\sqrt{3})(N_{2}+3N_{3})]/\\{2^{2/3}3[N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]^{1/3}\\}$
$+\frac{1}{62^{1/3}}(1+i\sqrt{3})\\{N_{4}+\surd[N_{4}^{2}+4(-N_{2}+3N_{3})^{3}]\\}^{1/3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(A3)$

where

$N_{1}=-f_{1}-f_{2}-f_{3}+k_{z}+2\sigma k_{z},$
$N_{2}=(f_{1}+f_{2}+f_{3}-k_{z}-2\sigma k_{z})^{2},$

$N_{3}=-|\kappa_{1}|^{2}-|\kappa_{2}|^{2}+f_{1}f_{2}+f_{1}f_{3}+f_{2}f_{3}-2\sigma
f_{1}k_{z}-f_{2}k_{z}-\sigma f_{2}k_{z}-f_{3}k_{z}-\sigma f_{3}k_{z}+2\sigma
k_{z}^{2}+\sigma^{2}k_{z}^{2}$

and

$N_{4}=-9|\kappa_{1}|^{2}f_{1}-9|\kappa_{2}|^{2}f_{1}-2f_{1}^{3}-9|\kappa_{1}|^{2}f_{2}+18|\kappa_{2}|^{2}f_{2}+3f_{1}^{2}f_{2}+3f_{1}f_{2}^{2}-2f_{2}^{3}$
$+18|\kappa_{1}|^{2}f_{3}-9|\kappa_{2}|^{2}f_{3}+3f_{1}^{2}f_{3}-12f_{1}f_{2}f_{3}+3f_{2}^{2}f_{3}+3f_{1}f_{3}^{2}+3f_{2}f_{3}^{2}-2f_{3}^{3}+9|\kappa_{1}|^{2}k_{z}+9|\kappa_{1}|^{2}k_{z}$
$-9|\kappa_{1}|^{2}\sigma k_{z}-9|\kappa_{2}|^{2}\sigma
k_{z}+6f_{1}^{2}k_{z}-6\sigma f_{1}^{2}k_{z}-6f_{1}f_{2}k_{z}+6\sigma
f_{1}f_{2}k_{z}$ $-3f_{2}^{2}k_{z}+3\sigma
f_{2}^{2}k_{z}-6f_{1}f_{3}k_{z}+6\sigma
f_{1}f_{3}k_{z}+12f_{2}f_{3}k_{z}-12\sigma f_{2}f_{3}k_{z}-3f_{3}^{2}k_{z}$
$+3\sigma f_{3}^{2}k_{z}-6f_{1}k_{z}^{2}+12\sigma
f_{1}k_{z}^{2}-6\sigma^{2}f_{1}k_{z}^{2}+3f_{2}k_{z}^{2}-6\sigma
f_{2}k_{z}^{2}+3\sigma^{2}f_{2}k_{z}^{2}+3f_{3}k_{z}^{2}$ $-6\sigma
f_{3}k_{z}^{2}+3\sigma^{2}f_{3}k_{z}^{2}+2k_{z}^{3}-6\sigma
k_{z}^{3}+6\sigma^{2}k_{z}^{3}-2\sigma^{3}k_{z}^{3}.$

It may be noted that the particular solutions given in Eq. (11) for the
special case $f_{2}$ = $f_{3}$ follows from the above.

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[label =0, label-width=0.9em,label-format=]tabbedEnum[](2)

# Decomposition of linear operators on pre-euclidean spaces by means of graphs

Hani Abdelwahab Hani Abdelwahab.
Mansoura University, Faculty of Science, Department of Mathematics (Egypt).
<EMAIL_ADDRESS>,  Elisabete Barreiro Elisabete Barreiro.
University of Coimbra, CMUC, Department of Mathematics, FCTUC, Largo D. Dinis
3000-143 Coimbra (Portugal<EMAIL_ADDRESS>,  Antonio J. Calderón Antonio J.
Calderón.
University of Cádiz, Department of Mathematics, Puerto Real (Spain).
<EMAIL_ADDRESS>and  José María Sánchez José María Sánchez.
University of Cádiz, Department of Mathematics, Puerto Real (Spain).
<EMAIL_ADDRESS>
###### Abstract.

In this work we study a linear operator $f$ on a pre-euclidean space
$\mathcal{V}$ by using properties of a corresponding graph. Given a basis
$\mathcal{B}$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as
an orthogonal direct sum of certain linear subspaces $\\{U_{i}\\}_{i\in I}$,
each one admitting a basis inherited from $\mathcal{B}$, in such way that
$f=\sum_{i\in I}f_{i}$, being each $f_{i}$ a linear operator satisfying
certain conditions respect with $U_{i}$. Considering new hypothesis, we assure
the existence of an isomorphism between the graphs associated to $f$ relative
to two different bases. We also study the minimality of $\mathcal{V}$ by using
the graph associated to $f$ relative to $\mathcal{B}$.

2020MSC: 47A65, 47B37, 05C90.

Keywords: Linear operators, pre-euclidean spaces, graph theory.

The second author was supported by the Centre for Mathematics of the
University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government
through FCT/MCTES. Third and fourth author is supported by the PCI of the UCA
‘Teoría de Lie y Teoría de Espacios de Banach’ and by the PAI with project
number FQM298. Third author also is supported by the project FEDER-
UCA18-107643 and by the Spanish project ‘Algebras no conmutativas y de caminos
de Leavitt. Algebras de evolución. Estructuras de Lie y variedades de
Einstein’.

##  1\. Introduction

This paper is motivated by the following problem: If we consider a family
$\\{f_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i}\\}_{i\in I}$ of linear operators,
where any $\mathcal{V}_{i}$ is a pre-euclidean space, then we can construct in
a natural way a new linear operator

$f:\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$\displaystyle\perp$}}
}}{\vbox{ \hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}\mathcal{V}_{i}\to\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}\mathcal{V}_{i}$

as $f=\sum_{i\in I}f_{i}$, where $\mathcal{V}:=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}\mathcal{V}_{i}$ is the pre-euclidean space defined by componentwise
operations. But, what about the converse? That is, if

$f:\mathcal{V}\to\mathcal{V}$

is a linear operator on a pre-euclidean space. Can we find a family
$\\{\mathcal{V}_{i}\\}_{i\in I}$ of pre-euclidean spaces and a family of
linear operators $\\{f_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i}\\}_{i\in I}$ in
such a way that

(1) $\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}\mathcal{V}_{i}\hskip 8.5359pt{\rm and}\hskip 8.5359ptf=\sum_{i\in I}f_{i}?$

The aim of the present work is to study this problem by giving a positive
answer. A pre-euclidean space is simply a linear space provided with a
bilinear form, hence our work covers a wide range of structures. We also note
that we use, as a tool for our study, techniques of graphs. This allows us to
get the decomposition (1) in an easy way, just by looking to the graph
associated to $f$ (and a fixed basis). Observe that this result gives us the
opportunity of recover a (possible) large linear operator
$f:\mathcal{V}\to\mathcal{V}$, from a family of easier linear operators
$f_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i}$, in a visual and computable way,
what we hope will be useful in any area dealing with linear operators on a
vector space endowed with a bilinear map. In recent years, the use of graphs
has increased in order to apply them in other areas [1, 2, 3, 4, 5].
Concretely, the applications of the graphs for the study of linear operators
and algebras [6, 7, 8, 9, 10, 11, 12]. We highlight that this topic is
currently very active as some recently published articles show [14, 15, 16,
22, 17, 18, 19, 20, 21, 13].

More in detail, for a linear operator $f:\mathcal{V}\to\mathcal{V}$ on a pre-
euclidean space $\mathcal{V}$ with fixed a basis $\mathcal{B}$, we obtain its
decomposition as an orthogonal direct sum of certain linear subspaces
$\\{U_{i}\\}_{i\in I}$, each one admitting a basis inherited from
$\mathcal{B}$, in such way that $f$ is decomposed as $f=\sum_{i\in I}f_{i}$,
being each $f_{i}$ a linear operator satisfying certain conditions relative to
$U_{i}$. Also, for the linear operator $f$, we present conditions in order to
guarantee the existence of an isomorphism between the graphs associated to $f$
relative to two different bases of $\mathcal{V}$. Finally, we analyze the
minimality property for $\mathcal{V}$ by using the graph associated to $f$
relative to $\mathcal{B}$.

The paper is organized as follows. Section 2 contains basic notions needed in
the sequel. In Section 3 we associate a graph $\Gamma(f,\mathcal{B})$ to any
linear operator $f$ defined on a pre-euclidean space
$(\mathcal{V},\langle\cdot,\cdot\rangle)$ with a fixed basis $\mathcal{B}$. In
addition, we introduce the notion of $f$-indecomposable in order to give a
characterization using the connectivity of $\Gamma(f,\mathcal{B})$. In Section
4 we give the definition of $f$-equivalence under which the graphs
$\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are isomorphic,
for two different bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ of
$\mathcal{V}$. Also, we relate this properties with the definition of
equivalent decomposition for $f$. In Section 5 we analyze the minimality
property for $\mathcal{V}$ by using $\Gamma(f,\mathcal{B})$, the graph
associated to $f$ relative to $\mathcal{B}$. Finally, we present an entire
conclusions section where we critically highlight our contribution, and
identifying strengths and weaknesses and proposing paths for future research.

##  2\. Basic definitions

Throughout this paper, $\mathbb{F}$ denotes an arbitrary field and all vector
spaces are assumed to be arbitrary dimensional and over base field
$\mathbb{F}$.

######  Definition 2.1.

A _pre-euclidean space_ is a pair $(\mathcal{V},\langle\cdot,\cdot\rangle)$,
where $\mathcal{V}$ is a $\mathbb{F}$-vector space and
$\langle\cdot,\cdot\rangle:\mathcal{V}\times\mathcal{V}\to\mathbb{F}$ is a
bilinear form.

######  Example 2.1.

$\mathbb{R}$ endowed with the bilinear form
$\langle\cdot,\cdot\rangle:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ given as
$\langle x,y\rangle:=\lambda xy$, for $x,y\in\mathbb{R}$ and fixed
$\lambda\in\mathbb{R}$, is a pre-euclidean space.

A pre-euclidean space over the field $\mathbb{R}$ endowed with an scalar
product is a pre-Hilbert space. Therefore, the results in this paper apply to
pre-Hilbert spaces. A pre-euclidean subspace of
$(\mathcal{V},\langle\cdot,\cdot\rangle)$ is a linear subspace $U$ of
$\mathcal{V}$ endowed with the bilinear form
$\langle\cdot,\cdot\rangle|_{U\times U}$. Additionally, given two pre-
euclidean spaces $(\mathcal{V},\langle\cdot,\cdot\rangle_{\mathcal{V}})$ and
$(\mathcal{W},\langle\cdot,\cdot\rangle_{\mathcal{W}})$, a morphism from
$\mathcal{V}$ to $\mathcal{W}$ is a linear map
$\phi:\mathcal{V}\to\mathcal{W}$ satisfying $\langle
x,y\rangle_{\mathcal{V}}=\langle\phi(x),\phi(y)\rangle_{\mathcal{W}}$ for
$x,y\in\mathcal{V}$. An isomorphism is a bijective morphism from $\mathcal{V}$
to $\mathcal{W}$. More, an automorphism is an isomorphism from $\mathcal{V}$
to itself.

######  Definition 2.2.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be a pre-euclidean space.

  1. i.

We say that two elements $x,y\in\mathcal{V}$ are orthogonal if $\langle
x,y\rangle=0$.

  2. ii.

The vector subspaces $U$ and $W$ of $\mathcal{V}$ are orthogonal if $\langle
u,w\rangle=0$ for $u\in U,w\in W$. In this case, we denote by $\langle
U,W\rangle=\\{0\\}$.

  3. iii.

$\mathcal{V}$ is an orthogonal direct sum of linear subspaces $U_{i}$ of
$\mathcal{V}$, with $i\in I$, denoted as

$\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}U_{i},$

if $\mathcal{V}$ decomposes as a direct sum $\mathcal{V}=\bigoplus_{i\in
I}U_{i}$ of linear subspaces $U_{i}$ such that $\langle
U_{i},U_{j}\rangle=\\{0\\}$ whenever $i\neq j$.

##  3\. Linear operator on a pre-euclidean space and graphs. Decomposition
Theorem

We recall that a (directed) graph is a pair $(V,E)$ where $V$ is a set of
vertices and $E\subset V\times V$ a set of (directed) edges connecting the
vertices.

######  Definition 3.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with fixed basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. The directed graph associated to $f$
relative to $\mathcal{B}$ is $\Gamma(f,\mathcal{B}):=(V,E)$, where
$V:=\mathcal{B}$ and

$E:=\Bigl{\\{}(e_{i},e_{j})\in V\times V:\\{\langle e_{i},e_{j}\rangle,\langle
e_{j},e_{i}\rangle\\}\neq\\{0\\}\mbox{ or
}f(e_{i})=\sum_{j}\lambda_{j}e_{j}\mbox{ for some
}0\neq\lambda_{j}\in\mathbb{F}\Bigr{\\}}.$

We say that $\Gamma(f,\mathcal{B})$ is the (directed) graph associated to $f$
relative to basis $\mathcal{B}$.

######  Example 3.1.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be the pre-euclidean space over
$\mathbb{R}$ with a fixed basis $\mathcal{B}=\\{e_{1},e_{2},\ldots,e_{5}\\}$
such that $\langle e_{2},e_{5}\rangle=7$ and the rest zero. Let
$f:\mathcal{V}\to\mathcal{V}$ be the linear operator defined as

$f(e_{1})=f(e_{3})=f(e_{5}):=e_{1}+2e_{3}+e_{5},\hskip
28.45274ptf(e_{2})=f(e_{4}):=-e_{4}.$

Then the associated graph $\Gamma(f,\mathcal{B})$ is:

$e_{1}$$e_{5}$$e_{3}$$e_{2}$$e_{4}$

######  Example 3.2.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be the pre-euclidean space over
$\mathbb{R}$ with basis $\mathcal{B}:=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ and
bilinear form defined as

$\langle v_{4},v_{3}\rangle=\langle v_{1},v_{5}\rangle:=1,\hskip
28.45274pt\langle v_{4},v_{2}\rangle=\langle v_{3},v_{5}\rangle:=3.$

Let $f:\mathcal{V}\to\mathcal{V}$ be the linear operator given as

$f(v_{1})=f(v_{3})=f(v_{2}):=2v_{1}+2v_{2}+v_{3},\hskip
28.45274ptf(v_{4})=v_{4}.$

So the graph $\Gamma(f,\mathcal{B})$ is:

$v_{2}$$v_{3}$$v_{1}$$v_{4}$$v_{5}$

Given two vertices $v_{i},v_{j}\in V$, an _undirected path_ from $v_{i}$ to
$v_{j}$ is a sequence of vertices $(v_{i_{1}},\dots,v_{i_{n}})$ with
$v_{i_{1}}=v_{i}$, $v_{i_{n}}=v_{j}$ and such that either
$(v_{i_{r}},v_{i_{r+1}})\in E$ or $(v_{i_{r+1}},v_{i_{r}})\in E$, for $1\leq
r\leq n-1$. We may introduce an equivalence relation in $V$: we say that
$v_{i}$ is related to $v_{j}$ in $V$, and denote $v_{i}\sim v_{j}$, if either
$v_{i}=v_{j}$ or there exists an undirected path from $v_{i}$ to $v_{j}$. In
this case, we assert that $v_{i}$ and $v_{j}$ are _connected_ and the
equivalence class of $v_{i}$, denoted by $[v_{i}]\in V/\sim$, corresponds to a
connected component $\mathcal{C}_{[v_{i}]}$ of the graph
$\Gamma(f,\mathcal{B})$. Therefore

(2) $\Gamma(f,\mathcal{B})=\dot{\bigcup_{[v_{i}]\in
V/\sim}}\mathcal{C}_{[v_{i}]}.$

We can also associate to any $\mathcal{C}_{[v_{i}]}$ the linear subspace

(3) ${\mathcal{V}}_{\mathcal{C}_{[v_{i}]}}:=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}}
}}}\displaylimits_{v_{j}\in[v_{i}]}\mathbb{F}v_{j}.$

######  Definition 3.2.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be a pre-euclidean space with
basis $\mathcal{B}$. A linear subspace $U$ of ${\mathcal{V}}$ admits a basis
$\mathcal{B}^{\prime}$ inherited from $\mathcal{B}$ if $\mathcal{B}^{\prime}$
is a basis of $U$ satisfying $\mathcal{B}^{\prime}\subset\mathcal{B}$.

######  Definition 3.3.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. The space ${\mathcal{V}}$ is
$f$-decomposable with respect to $\mathcal{B}$ if ${\mathcal{V}}=U_{1}\perp
U_{2}$, being $U_{1},U_{2}$ non-zero linear subspaces admitting each one a
basis inherited from $\mathcal{B}$ and $f(U_{1})\subset U_{1}$,
$f(U_{2})\subset U_{2}$. Otherwise, ${\mathcal{V}}$ is said $f$-indecomposable
with respect to $\mathcal{B}$.

######  Example 3.3.

The pre-euclidean space over $\mathbb{R}$ of Example 3.1 is $f$-indecomposable
with respect $\mathcal{B}$

######  Example 3.4.

Let $\mathcal{V}$ be the pre-euclidean space defined by: the $5$-dimensional
$\mathbb{C}$-vector space with basis
$\mathcal{B}:=\\{e_{1},e_{2},e_{3},e_{4},e_{5}\\}$ and bilinear form defined
as

$\langle e_{1},e_{3}\rangle:=4i,\hskip 56.9055pt\langle
e_{4},e_{5}\rangle:=2-11i,$

and the rest zero. We consider the linear operator
$f:\mathcal{V}\to\mathcal{V}$ given as

$f(e_{1}):=2e_{1}-e_{2},\hskip 56.9055ptf(e_{2}):=e_{3},$

and zero on the rest. Then by denoting $U_{1},U_{2}$ the $\mathbb{C}$-linear
subspaces of $\mathcal{V}$ with bases $\\{e_{1},e_{2},e_{3}\\}$,
$\\{e_{4},e_{5}\\}$, respectively, we easily see that

${\mathcal{V}}=U_{1}\perp U_{2}$

and ${\mathcal{V}}$ is $f$-decomposable with respect to $\mathcal{B}$. The
associated graph $\Gamma(f,\mathcal{B})$ is:

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$

A graph $(V,E)$ is _connected_ if any two vertices are connected.
Equivalently, a graph is connected if and only if for every partition of its
vertices into two non-empty sets, there is an edge with an endpoint in each
set.

######  Theorem 3.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. Then the following statements are
equivalent.

  1. i.

The graph $\Gamma(f,\mathcal{B})$ is connected.

  2. ii.

$\mathcal{V}$ is $f$-indecomposable with respect to $\mathcal{B}$.

###### Proof.

First we suppose that the graph $\Gamma(f,\mathcal{B})$ is connected. Let us
assume that $\mathcal{V}$ is $f$-decomposable with respect to $\mathcal{B}$.
So $\mathcal{V}$ is the orthogonal direct sum

$\mathcal{V}=U_{1}\perp U_{2}$

of two linear subspaces $U_{1}$ and $U_{2}$ admitting each one a basis
$\mathcal{B}_{1}:=\\{e_{j}:j\in J\\}$ and $\mathcal{B}_{2}:=\\{e_{k}:k\in
K\\}$, respectively, inherited from $\mathcal{B}$ such that $f(U_{1})\subset
U_{1}$ and $f(U_{2})\subset U_{2}$. Hence
$\mathcal{B}=\mathcal{B}_{1}\;\dot{\cup}\;\mathcal{B}_{2}.$ Fix some
$e_{j}\in\mathcal{B}_{1}$ and $e_{k}\in\mathcal{B}_{2}$. Since the graph
$\Gamma(f,\mathcal{B})$ is connected, it follows that $e_{j}$ is connected to
$e_{k}$. So there exists an undirected path

$(e_{j},v_{i_{2}},\dots,v_{i_{n-1}},e_{k})$

from $e_{j}$ to $e_{k}$. From here, there are
$e^{\prime}=v_{i_{s}}\in\mathcal{B}_{1}$ and
$e^{\prime\prime}=v_{i_{s+1}}\in\mathcal{B}_{2}$ such that either
$(e^{\prime},e^{\prime\prime})\in E$ or $(e^{\prime\prime},e^{\prime})\in E$.
Since $\langle e^{\prime},e^{\prime\prime}\rangle=\langle
e^{\prime\prime},e^{\prime}\rangle=0$, we have either
$P_{\mathbb{F}e^{\prime\prime}}\bigl{(}f(e^{\prime})\bigr{)}\neq 0$ or
$P_{\mathbb{F}e^{\prime}}\bigl{(}f(e^{\prime\prime})\bigr{)}\neq 0$, where
$P_{U}:\mathcal{V}\to U$ is the projection of $\mathcal{V}$ onto the linear
subspace $U$. Hence, we have either $f(U_{1})\not\subset U_{1}$ or
$f(U_{2})\not\subset U_{2}$, in both cases it is a contradiction. Therefore,
$\mathcal{V}$ is $f$-indecomposable with respect to $\mathcal{B}$.

Conversely, let us suppose that $\mathcal{V}$ is $f$-indecomposable with
respect to $\mathcal{B}$ and $\Gamma(f,\mathcal{B})$ is not connected. Then
there exists a partition
$\mathcal{B}=\mathcal{B}_{1}\;\dot{\cup}\;\mathcal{B}_{2}$ such that both
$(x,y)$ and $(y,x)$ are not in $E$, for any $x\in\mathcal{B}_{1}$ and
$y\in\mathcal{B}_{2}$. Set $U_{1}:=\bigoplus_{x\in\mathcal{B}_{1}}\mathbb{F}x$
and $U_{2}:=\bigoplus_{y\in\mathcal{B}_{2}}\mathbb{F}y$. Then we have $\langle
x,y\rangle=\langle y,x\rangle=0$ for any $x\in\mathcal{B}_{1}$ and
$y\in\mathcal{B}_{2}$. Moreover, $f(U_{1})\subset U_{1}$ and $f(U_{2})\subset
U_{2}$. So $\mathcal{V}$ is the orthogonal direct sum

$\mathcal{V}=U_{1}\perp U_{2},$

of two linear subspaces $U_{1}$ and $U_{2}$ admitting each one a basis
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, respectively, inherited from
$\mathcal{B}$. Thus, $\mathcal{V}$ is $f$-decomposable with respect to
$\mathcal{B}$, which is a contradiction. ∎

######  Corollary 3.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. Then, for each $[v_{i}]\in V/\sim$, the
linear subspace
$\mathcal{V}_{\mathcal{C}_{[v_{i}]}}:=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}}
}}}\displaylimits_{v_{j}\in[v_{i}]}\mathbb{F}v_{j}$ of $\mathcal{V}$ is
$f$-indecomposable with respect to $[v_{i}]$.

We illustrate the result with this simple example.

######  Example 3.5.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be the pre-euclidean space over
$\mathbb{R}$ with a fixed basis $\mathcal{B}=\\{e_{1},e_{2},e_{3},e_{4}\\}$
such that $\langle e_{1},e_{3}\rangle=-5$, $\langle e_{2},e_{4}\rangle=1$, and
zero on the rest. Let $f:\mathcal{V}\to\mathcal{V}$ be the linear operator
defined as

$f(e_{1})=f(e_{3}):=-2e_{3},\hskip 28.45274ptf(e_{2})=f(e_{4}):=5e_{2}+e_{4}.$

Then the associated graph $\Gamma(f,\mathcal{B})$ is:

$e_{1}$$e_{3}$$e_{2}$$e_{4}$

We have ${\mathcal{V}}=U_{1}\perp U_{2}$ where $U_{1},U_{2}$ are the subspaces
with bases $\\{e_{1},e_{3}\\}$ and $\\{e_{2},e_{4}\\}$, respectively. Since
the graph $\Gamma(f,\mathcal{B})$ is not connected, we conclude $\mathcal{V}$
is $f$-decomposable with respect to $\mathcal{B}$. However, $U_{1}$ and
$U_{2}$ are $f$-indecomposable with respect to $\\{e_{1},e_{3}\\}$ and
$\\{e_{2},e_{4}\\}$, respectively.

######  Theorem 3.2.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$. Then for a fixed basis
$\mathcal{B}:=\\{e_{j}\\}_{j\in J}$ of ${\mathcal{V}}$ it holds that

${\mathcal{V}}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}U_{i},$

being each $U_{i}$ a linear subspace of ${\mathcal{V}}$ admitting
$\mathcal{B}_{[i]}:=\\{e_{j}:e_{j}\in[e_{i}]\\}$ as a basis inherited from
$\mathcal{B}$. Also, we have

$f=\sum_{i\in I}f_{i},$

being each $f_{i}$ a linear operator on the pre-euclidean space
$(\mathcal{V},\langle\cdot,\cdot\rangle)$, for $i\in I$, such that

$f_{i}|_{U_{i}}=f|_{U_{i}},\hskip 56.9055ptf_{i}(U_{i})\subset U_{i},\hskip
56.9055ptf_{i}(\perp_{i\neq j}U_{j})=0.$

Further, for each $i\in I$, $U_{i}$ is $f_{i}$-indecomposable with respect to
$\mathcal{B}_{[i]}$.

###### Proof.

Let $V/\sim=\\{[e_{i}]\\}_{i\in I}$. Then, from Equations (2) and (3), we can
assert that $\mathcal{V}:=\bigoplus_{i\in I}U_{i}$ is the direct sum of the
family of linear subspaces $U_{i}:=\mathcal{V}_{\mathcal{C}_{[e_{i}]}}$ and
$\mathcal{B}_{[i]}=\\{e_{j}:e_{j}\in[e_{i}]\\}$ is the basis for $U_{i}$, with
$i\in I$. So if $U_{i}\neq U_{j}$ then $[e_{i}]\neq[e_{j}]$ and hence
$\\{\langle x,y\rangle,\langle y,x\rangle\\}=\\{0\\}$ for any $x\in[e_{i}]$,
$y\in[e_{j}]$. From here, it follows that $\langle U_{i},U_{j}\rangle=\langle
U_{j},U_{i}\rangle=0$ for any $i\neq j$. That is,
$\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}U_{i}$.

Assume now that $P_{U_{j}}\left(f(U_{i})\right)\neq\\{0\\}$ for $i\neq j$,
where $P_{U_{j}}:\mathcal{V}\to U_{j}$ is the projection of $\mathcal{V}$ onto
$U_{j}$. Then there exist $e^{\prime}\in[e_{i}]$ and
$e^{\prime\prime}\in[v_{j}]$ such that $P_{U_{j}}(f(e^{\prime}))=\lambda
e^{\prime\prime}+u$ with $\lambda\in\mathbb{F}\setminus\\{0\\}$ and
$u\in\bigoplus_{e\neq e^{\prime\prime}\in[e_{j}]}\mathbb{F}e\subset U_{j}$.
Since $\lambda\neq 0$, we have $(e^{\prime},e^{\prime\prime})\in E$ and
therefore $e^{\prime}\sim e^{\prime\prime}$, i.e.
$[e^{\prime}]=[e^{\prime\prime}]$. As $[e_{i}]=[e^{\prime}]$ and
$[e_{j}]=[e^{\prime\prime}]$, we have $[e_{i}]=[e_{j}]$ and so $U_{i}=U_{j}$,
a contradiction. Thus $P_{\perp_{i\neq j}U_{j}}\bigl{(}f(U_{i})\bigr{)}=0$ and
we conclude that $f(U_{i})\subset U_{i}$. Consequently, each linear subspace
$U_{i}$ of $\mathcal{V}$ admits a basis $\mathcal{B}_{[i]}$ inherited from
$\mathcal{B}$.

Since $\mathcal{V}=\perp_{i\in I}U_{i}$, for each $i\in I$ we define the
linear operator $f_{i}:\mathcal{V}\to\mathcal{V}$ as $f_{i}(U_{i}):=f(U_{i})$
and $f_{i}(\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\neq
j}U_{j}):=\\{0\\}$, so $f=\sum_{i\in I}f_{i}$.

Now, let us show that each $U_{i}$ is $f_{i}$-indecomposable with respect to
$\mathcal{B}_{[i]}$. We assume that

$U_{i}=U_{1}^{\prime}\perp U_{2}^{\prime},$

where $U_{1}^{\prime}$ and $U_{2}^{\prime}$ are non-zero linear subspaces of
$U_{i}$ admitting the $f_{i}$-basis $\mathcal{B}_{1}:=\\{e_{j}:j\in J\\}$ and
$\mathcal{B}_{2}:=\\{e_{k}:k\in K\\}$ inherited from $\mathcal{B}_{[i]}$,
respectively. That is,

$\mathcal{B}_{[i]}=\mathcal{B}_{1}\;\dot{\cup}\;\mathcal{B}_{2}.$

Fix some $e_{j}\in\mathcal{B}_{1}$ and $e_{k}\in\mathcal{B}_{2}$. Since
$e_{j}$ is connected to $e_{k}$, there exists an undirected path

$(e_{j},v_{i_{2}},\dots,v_{i_{n-1}},e_{k})$

from $e_{j}$ to $e_{k}$. From here, there are
$e^{\prime}=v_{i_{s}}\in\mathcal{B}_{1}$ and
$e^{\prime\prime}=v_{i_{s+1}}\in\mathcal{B}_{2}$ such that either
$(e^{\prime},e^{\prime\prime})\in E$ or $(e^{\prime\prime},e^{\prime})\in E$.
Since $\langle e^{\prime},e^{\prime\prime}\rangle=\langle
e^{\prime\prime},e^{\prime}\rangle=0$, we have either
$P_{\mathbb{F}e^{\prime\prime}}(f_{i}(e^{\prime}))\neq 0$ or
$P_{\mathbb{F}e^{\prime}}(f_{i}(e^{\prime\prime}))\neq 0$. Therefore, we have
either $f_{i}(U_{1}^{\prime})\not\subset U_{1}^{\prime}$ or
$f_{i}(U_{2}^{\prime})\not\subset U_{2}^{\prime}$. In both cases, it is a
contradiction and the proof is completed. ∎

######  Example 3.6.

Let $\mathcal{V}$ the $6$-dimensional $\mathbb{F}$-vector space with basis
$\mathcal{B}:=\\{e_{1},e_{2},\dots,e_{6}\\}$ and bilinear form defined as

$\langle e_{1},e_{2}\rangle:=\alpha,\hskip 56.9055pt\langle
e_{5},e_{6}\rangle:=\beta,$

and the rest zero, with $\alpha,\beta\in\mathbb{F}\setminus\\{0\\}$. We
consider the linear operator $f:\mathcal{V}\to\mathcal{V}$ given as

$f(e_{1}):=\alpha e_{1},\hskip 56.9055ptf(e_{3}):=\beta e_{2},\hskip
56.9055ptf(e_{4}):=-\beta e_{4},\hskip 56.9055ptf(e_{5}):=-\alpha e_{5}+\alpha
e_{6},$

and zero on the rest, with $\alpha,\beta$ the same previous scalars. By
denoting $U_{1},U_{2},U_{3}$ the linear subspaces of $\mathcal{V}$ with bases
$\mathcal{B}_{[1]}:=\\{e_{1},e_{2},e_{3}\\}$,
$\mathcal{B}_{[2]}:=\\{e_{4}\\}$, $\mathcal{B}_{[3]}:=\\{e_{5},e_{6}\\}$,
respectively, we have $\mathcal{V}$ is $f$-decomposable with respect to
$\mathcal{B}$, since

${\mathcal{V}}=U_{1}\perp U_{2}\perp U_{3}.$

The associated graph $\Gamma(f,\mathcal{B})$ is:

$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$$e_{6}$

Also, for $i\in\\{1,2,3\\}$, we define $f_{i}:{\mathcal{V}}\to{\mathcal{V}}$
as $f_{1}(e_{1}):=\alpha e_{1}$, $f_{1}(e_{3}):=\beta e_{2}$,
$f_{2}(e_{4}):=-\beta e_{4}$, $f_{3}(e_{5}):=-\alpha e_{5}+\alpha e_{6}$ and
zero on the rest (being $\alpha,\beta$ the same previous non-zero scalars). We
get that

$f=f_{1}+f_{2}+f_{3}$

satisfies the condition of Theorem 3.2, then $U_{i}$ is $f_{i}$-indecomposable
with respect to $\mathcal{B}_{[i]}$, for $i\in\\{1,2,3\\}$. So obviously,
${\mathcal{V}}$ is $f$-decomposable with respect to $\mathcal{B}$.

To identify the components of the decomposition given in Theorem 3.2 we only
need to focus on the connected components of the associated graph.

##  4\. Relating the graphs given by different choices of bases

In general, for a linear operator $f:\mathcal{V}\to\mathcal{V}$ on a pre-
euclidean space $(\mathcal{V},\langle\cdot,\cdot\rangle)$, two different bases
of $\mathcal{V}$ determine associated graphs not isomorphic, which can give
rise to different decomposition of $f$ as in Theorem 3.2. We recall that two
graphs $(V,E)$ and $(V^{\prime},E^{\prime})$ are isomorphic if there exists a
bijection $\phi:V\to V^{\prime}$ such that $(v_{i},v_{j})\in E$ if and only if
$(\phi(v_{i}),\phi(v_{j}))\in E^{\prime}$. For instance, we show the next
example.

######  Example 4.1.

For the linear operator and the pre-euclidean space of Example 3.2, if we
consider $w_{1}:=v_{1}+v_{2}$, $w_{2}:=v_{1}-v_{2}$, $w_{3}:=v_{4}+v_{5}$,
$w_{4}:=v_{4}-v_{5}$ and $w_{5}:=v_{3}$, for the basis
$\mathcal{B}^{\prime}:=\\{w_{1},w_{2},w_{3},w_{4},w_{5}\\}$ we get

$\langle w_{1},w_{3}\rangle=-\langle w_{1},w_{4}\rangle=\langle
w_{2},w_{3}\rangle=-\langle w_{2},w_{4}\rangle=\langle
w_{3},w_{5}\rangle=\langle w_{4},w_{5}\rangle=1,$ $\langle
w_{3},w_{1}\rangle=-\langle w_{3},w_{2}\rangle=\langle
w_{4},w_{1}\rangle=-\langle w_{4},w_{2}\rangle=\langle
w_{5},w_{3}\rangle=-\langle w_{5},w_{4}\rangle=3,$

and also

$f(w_{1})=4w_{1}+2w_{5},\hskip
28.45274ptf(w_{3})=f(w_{4})=\frac{1}{2}w_{3}+\frac{1}{2}w_{4},\hskip
28.45274ptf(w_{5})=2w_{1}+w_{5}.$

So we obtain the associated graph $\Gamma(f,\mathcal{B}^{\prime})$ as:

$w_{1}$$w_{5}$$w_{2}$$w_{3}$$w_{4}$

Clearly, $\Gamma(f,\mathcal{B}^{\prime})$ is not isomorphic to the associated
graph $\Gamma(f,\mathcal{B})$ stated in Example 3.2.

Next we give a condition under which the graphs associated to a linear
operator $f:\mathcal{V}\to\mathcal{V}$ on a pre-euclidean space
$(\mathcal{V},\langle\cdot,\cdot\rangle)$, performed by two different bases,
are isomorphic. As a consequence, we establish a sufficient condition under
which two decomposition of $f$, induced by two different bases, are
equivalent.

######  Definition 4.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$. Two bases
$\mathcal{B}=\\{v_{i}\\}_{i\in I}$ and $\mathcal{B}^{\prime}=\\{w_{i}\\}_{i\in
I}$ of $\mathcal{V}$ are $f$-equivalent if there exists an automorphism
$\phi:\mathcal{V}\to\mathcal{V}$ satisfying
$\phi(\mathcal{B})=\mathcal{B}^{\prime}$ and $\phi\circ f=f\circ\phi$.

######  Lemma 4.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. Consider two bases $\mathcal{B}$ and
$\mathcal{B}^{\prime}$ of $\mathcal{V}$. If $\mathcal{B}$ and
$\mathcal{B}^{\prime}$ are $f$-equivalent bases then the associated graphs
$\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are isomorphic.

###### Proof.

Let us suppose that $\mathcal{B}=\\{v_{i}\\}_{i\in
I},\mathcal{B}^{\prime}=\\{w_{i}\\}_{i\in I}$ are two $f$-equivalent bases of
$\mathcal{V}$. Then there exists an automorphism
$\phi:\mathcal{V}\to\mathcal{V}$ satisfying $f\circ\phi=\phi\circ f$ and

(4) $\langle x,y\rangle=\langle\phi(x),\phi(y)\rangle$

for $x,y\in\mathcal{V}$, in such way that for every $v_{i}\in\mathcal{B}$
there exists an unique $w_{j_{i}}\in\mathcal{B}^{\prime}$ verifying
$\phi(v_{i})=w_{j_{i}}$.

Let us denote by $(V,E)$ and $(V^{\prime},E^{\prime})$ the set of vertices and
edges of $\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$,
respectively. Taking into account that $V=\mathcal{B}$ and
$V^{\prime}=\mathcal{B}^{\prime}$, and the fact
$\phi(\mathcal{B})=\mathcal{B}^{\prime}$, we have that $\phi$ defines a
bijection from $V$ to $V^{\prime}$. Given $v_{i},v_{k}\in V$, we want to show
that $(v_{i},v_{k})\in E$ if and only if
$(\phi(v_{i}),\phi(v_{k}))=(w_{j_{i}},w_{j_{k}})\in E^{\prime}$. Suppose that
$(v_{i},v_{k})\in E$, thus either $\\{\langle v_{i},v_{k}\rangle,\langle
v_{k},v_{i}\rangle\\}\neq\\{0\\}$ or $f(v_{i})=\sum_{k}\lambda_{k}v_{k}\mbox{
for some }0\neq\lambda_{k}\in\mathbb{F}$. If $\\{\langle
v_{i},v_{k}\rangle,\langle v_{k},v_{i}\rangle\\}\neq\\{0\\}$ then by Equation
(4) we have $\\{\langle w_{j_{i}},w_{j_{k}}\rangle,\langle
w_{j_{k}},w_{j_{i}}\rangle\\}\neq\\{0\\}$. If
$f(v_{i})=\sum_{k}\lambda_{k}v_{k}\mbox{ for some
}0\neq\lambda_{k}\in\mathbb{F}$, applying $\phi$ to this relation we get that
$f(w_{j_{i}})=f(\phi(v_{i}))=\phi(f(v_{i}))=\phi(\sum_{k}\lambda_{k}v_{k})=\sum_{k}\lambda_{k}\phi(v_{k})=\sum_{k}\lambda_{k}w_{j_{k}}\mbox{
for some }0\neq\lambda_{k}\in\mathbb{F}$. Thus, $(w_{j_{i}},w_{j_{k}})\in
E^{\prime}$. The same argument using $\phi^{-1}$ shows that if
$(w_{j_{i}},w_{j_{k}})\in E^{\prime}$ then $(v_{i},v_{k})\in E$, because also
$\phi^{-1}\circ f=f\circ\phi^{-1}$. This fact concludes the proof that
$\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are isomorphic
via $\phi$. ∎

The following concept is borrowed from the theory of graded algebras (see for
instance [23]).

######  Definition 4.2.

Let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be a pre-euclidean space and let

$\Upsilon:=\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{i\in
I}\mathcal{V}_{i}\hskip 11.38092pt\mbox{\rm and}\hskip
11.38092pt\Upsilon^{\prime}:=\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{j\in
J}\mathcal{V}^{\prime}_{j}$

be two decomposition of $\mathcal{V}$ as an orthogonal direct sum of linear
subspaces. It is said that $\Upsilon$ and $\Upsilon^{\prime}$ are equivalent
if there exists an automorphism $\phi:\mathcal{V}\to\mathcal{V}$, and a
bijection $\sigma:I\to J$ such that
$\phi(\mathcal{V}_{i})=\mathcal{V}^{\prime}_{\sigma(i)}$ for all $i\in I$.

######  Theorem 4.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$. Then for two bases
$\mathcal{B}:=\\{v_{i}\\}_{i\in I}$ and
$\mathcal{B}^{\prime}:=\\{v_{j}^{\prime}\\}_{j\in J}$ of $\mathcal{V}$
consider the following assertions:

  * i.

The bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are $f$-equivalent.

  * ii.

The graphs $\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are
isomorphic.

  * iii.

The decomposition of the linear operator $f:\mathcal{V}\to\mathcal{V}$ with
respect to $\mathcal{B}$

$\Upsilon:=\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{[v_{i}]\in
V/\sim}\mathcal{V}_{\mathcal{C}_{[v_{i}]}},$

given by

$f=\sum_{i\in I}f_{i}$

with
$f_{i}|_{\mathcal{V}_{\mathcal{C}_{[v_{i}]}}}=f|_{\mathcal{V}_{\mathcal{C}_{[v_{i}]}}}$,
$f_{i}(\mathcal{V}_{\mathcal{C}_{[v_{i}]}})\subset\mathcal{V}_{\mathcal{C}_{[v_{i}]}}$,
$f_{i}(\perp_{i\neq k}\mathcal{V}_{\mathcal{C}_{[v_{k}]}})=0,$ and the
decomposition of $f$ with respect to $\mathcal{B}^{\prime}$

$\Upsilon^{\prime}:=\mathcal{V}=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}}
}}}\displaylimits_{[v^{\prime}_{j}]\in
V^{\prime}/\sim}\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{j}]}},$

performed by

$f=\sum_{j\in J}f_{j}^{\prime}$

with
$f^{\prime}_{j}|_{\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{j}]}}}=f|_{\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{j}]}}}$,
$f^{\prime}_{j}(\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{j}]}})\subset\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{j}]}}$,
$f^{\prime}_{j}(\perp_{i\neq
k}\mathcal{V}_{\mathcal{C}^{\prime}_{[v^{\prime}_{k}]}})=0,$ are equivalent.

Then i. implies ii. and iii.

###### Proof.

The implication from i. to ii. was proved in Lemma 4.1. Let us prove the
implication from i. to iii. Suppose that $\phi:\mathcal{V}\to\mathcal{V}$ is
an automorphism satisfying $\phi(\mathcal{B})=\mathcal{B}^{\prime}$ and
$\phi\circ f=f\circ\phi$. By the implication from i. to ii., we know that
$\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are isomorphic
via $\phi$ and thus $\phi([v])=[\phi(v)]$, for all $v\in V=\mathcal{B}$. It
follows that
$\phi(\mathcal{V}_{\mathcal{C}_{[v]}})=\mathcal{V}_{\mathcal{C}_{[\phi(v)]}}$,
for all $[v]\in V/\sim$, which proves that the decomposition of $\mathcal{V}$
corresponding to $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are equivalent. ∎

######  Remark 4.1.

In general, the implication ii. to i. of Theorem 4.1 (as well the converse of
Lemma 4.1) is not valid. That is, the fact that associated graphs respect two
bases are isomorphic does not imply that these two bases are $f$-equivalent.
Let $\mathcal{V}$ be a pre-euclidean space with basis
$\mathcal{B}:=\\{v_{1},v_{2},v_{3}\\}$ and endowed with a bilinear form
defined as

$\langle v_{1},v_{1}\rangle=\langle v_{2},v_{2}\rangle=\langle
v_{3},v_{3}\rangle=\langle v_{1},v_{2}\rangle:=1$

and zero on the rest. By denoting $w_{1}:=v_{1}+v_{2}$, $w_{2}:=v_{1}-v_{2}$
and $w_{3}:=v_{3}$ we consider the basis
$\mathcal{B}^{\prime}:=\\{w_{1},w_{2},w_{3}\\}$, so we obtain

$\langle w_{1},w_{1}\rangle=3,\hskip 28.45274pt\langle
w_{2},w_{2}\rangle=\langle w_{3},w_{3}\rangle=\langle
w_{2},w_{1}\rangle=1,\hskip 28.45274pt\langle w_{1},w_{2}\rangle=-1.$

Therefore, for a zero linear operator $f$, the associated graphs
$\Gamma(f,\mathcal{B})$ and $\Gamma(f,\mathcal{B}^{\prime})$ are isomorphic:

$v_{3}$$v_{1}$$v_{2}$$w_{3}$$w_{1}$$w_{2}$

However, $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are not $f$-equivalent.
Indeed, if there exists an isomorphism $\phi:\mathcal{V}\to\mathcal{V}$ such
that $\phi(\mathcal{B})=\mathcal{B}^{\prime}$ we get, for instance,

$\phi(v_{1}):=w_{1},\hskip 28.45274pt\phi(v_{2}):=w_{3},\hskip
28.45274pt\phi(v_{3}):=w_{2},$

but $0=\langle
v_{1},v_{3}\rangle\neq\langle\phi(v_{1}),\phi(v_{3})\rangle=\langle
w_{1},w_{2}\rangle=-1.$

##  5\. Characterization of the minimality and weak symmetry

Let $(V,E)$ be a graph. Given $v_{i},v_{j}\in V$, we say that a directed path
from $v_{i}$ to $v_{j}$ is a sequence of vertices
$(v_{i_{1}},\dots,v_{i_{n}})$ satisfying $v_{i_{1}}=v_{i}$, $v_{i_{n}}=v_{j}$
and such that $(v_{i_{r}},v_{i_{r+1}})\in E$, for $1\leq r\leq n-1$. We also
say that $(V,E)$ is symmetric if $(v_{i},v_{j})\in E$ for all
$(v_{j},v_{i})\in E$. So we present the next (weaker) concept as follows.

######  Definition 5.1.

A graph $(V,E)$ is weakly symmetric if for any $(e_{j},e_{i})\in E$ there
exists a directed path from $e_{i}$ to $e_{j}$.

Of course, every symmetric graph is weakly symmetric.

######  Example 5.1.

The following graphs are weakly symmetric:

######  Definition 5.2.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis $\mathcal{B}$. We
say $\mathcal{V}$ is minimal if the unique pre-euclidean subspaces $U$
admitting a inherited basis from $\mathcal{B}$ such that $f(U)\subset U$ are
$\\{0\\},\mathcal{V}$.

######  Theorem 5.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. If $\mathcal{V}$ is minimal, then the
associated graph $\Gamma(f,\mathcal{B})$ to $f$ relative to $\mathcal{B}$ is
weakly symmetric.

###### Proof.

If $\mathcal{V}$ is minimal we have that ${\mathcal{V}}$ is the unique non-
zero pre-euclidean subspace that satisfies $f(\mathcal{V})\subset\mathcal{V}$.
Let $\Gamma(f,\mathcal{B}):=(V,E)$ be the associated graph and take some
$(e_{i},e_{j})\in E$. Therefore, either $\langle e_{i},e_{j}\rangle\neq 0$ or
$\langle e_{j},e_{i}\rangle\neq 0$ or $f(e_{i})=\sum_{j}\lambda_{j}e_{j}$ for
some $0\neq\lambda_{j}\in\mathbb{F}$. The first two cases imply
$(e_{j},e_{i})\in E$. In the last case, we have $f\neq 0$. Let us now define

$\mathcal{B}_{j}:=\\{e_{k}\in\mathcal{B}:e_{k}=e_{j}\mbox{ or there exists a
directed path from }e_{j}\mbox{ to }e_{k}\\}.$

Since $e_{j}\in\mathcal{B}_{j}$, we have $\mathcal{B}_{j}\neq\emptyset$. So
let $U$ be the space spanned by $\mathcal{B}_{j}$. Let
$e_{t}\in\mathcal{B}_{j}$ and $f(e_{t})=\sum_{m}\lambda_{m}e_{m}$. If
$\lambda_{m}\neq 0$ we get $(e_{t},e_{m})\in E$ and hence there exists a
directed path from $e_{j}$ to $e_{m}$:

$e_{j}$$\cdots$$\cdots$$\cdots$$e_{t}$$e_{m}$

So it implies $e_{m}\in\mathcal{B}_{j}$ and therefore $f(U)\subset U$. Since
$\mathcal{B}_{j}\neq\emptyset$ and ${\mathcal{V}}$ is minimal we have
$U=\mathcal{V}$, so we conclude $\mathcal{B}_{j}=\mathcal{B}$ and then
$e_{i}\in\mathcal{B}_{j}$. Thus there exists a directed path from $e_{j}$ to
$e_{i}$ as required. ∎

######  Corollary 5.1.

Let $f:\mathcal{V}\to\mathcal{V}$ be a linear operator on a pre-euclidean
space $(\mathcal{V},\langle\cdot,\cdot\rangle)$ with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$. If $\mathcal{V}$ is minimal, then the
associated graph $\Gamma(f,\mathcal{B})$ to $f$ relative to $\mathcal{B}$ is
connected.

###### Proof.

This is an immediate consequence of Corollary 3.1, Theorem 3.1 and Theorem
3.2. ∎

######  Remark 5.1.

In general, the converse of the previous results are not valid. As
counterexample, let $(\mathcal{V},\langle\cdot,\cdot\rangle)$ be the pre-
euclidean space over $\mathbb{R}$ with basis
$\mathcal{B}:=\\{e_{1},e_{2},e_{3},e_{4}\\}$ and bilinear form given as

$\langle e_{1},e_{2}\rangle=\langle e_{2},e_{3}\rangle=\langle
e_{3},e_{4}\rangle=\langle e_{4},e_{1}\rangle:=1.$

Let $f:\mathcal{V}\to\mathcal{V}$ be the linear operator defined as
$f(e_{i}):=e_{i}$ for $i\in\\{1,2,3,4\\}$. Then $\mathcal{V}$ is not minimal
since, for instance, the pre-euclidean subspace $U$ with inherited basis
$\\{e_{1}\\}$ from $\mathcal{B}$ satisfies $f(U)\subset U$. However, the
associated graph $\Gamma(f,\mathcal{B})$ is clearly connected and weakly
symmetric:

$e_{1}$$e_{2}$$e_{3}$$e_{4}$

##  6\. Conclusion

We were motivated by the following situation: If we consider a family of
linear operators

$\\{f_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i}\\}_{i\in I},$

where any $V_{i}$ is a pre-euclidean space, then we can construct in a natural
way a new linear operator

$f:\bigoplus_{i\in I}\mathcal{V}_{i}\to\bigoplus_{i\in I}\mathcal{V}_{i}$

as $f=\sum_{i\in I}f_{i}$, where $\mathcal{V}:=\bigoplus_{i\in
I}\mathcal{V}_{i}$ is the pre-euclidean space defined by componentwise
operations.

In this paper, our purpose has been to study the converse problem, and we have
given a positive answer by proving in Theorem 3.2 that given a linear operator
$f:\mathcal{V}\to\mathcal{V}$ on a pre-euclidean space, it is possible to find
a family of pre-euclidean spaces $\\{\mathcal{V}_{i}\\}_{i\in I}$, and a
family of linear operators
$\\{f_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i}\\}_{i\in I}$ in such a way that
$\mathcal{V}=\perp_{i\in I}\mathcal{V}_{i}$ and $f=\sum_{i\in I}f_{i}$. In
order to approach our question, we have used techniques of graphs. This allows
us to obtain the above decomposition of the pre-euclidean space $\mathcal{V}$
and the linear operator $f$ in a very practical way, just by looking to the
graph $\Gamma(F,\mathcal{B})$ associated to $f$ (and a fixed basis
$\mathcal{B}$). Also, the minimality of our structure is characterized in
Theorem 5.1, which says that, for a linear operator
$f:\mathcal{V}\to\mathcal{V}$ on a minimal pre-euclidean space
$(\mathcal{V},\langle\cdot,\cdot\rangle)$ (with basis
$\mathcal{B}=\\{e_{i}\\}_{i\in I}$), then the associated graph
$\Gamma(f,\mathcal{B})$ to $f$ relative to $\mathcal{B}$ is weakly symmetric.

Finally, we would like to note that future research in this topic are to
generalize this result for different classes of operators (not necessarily
linear operators) and, of course, consider operators on structures different
to pre-euclidean spaces (for instance Banach spaces).

Acknowledgment. The authors would like to thank the reviewers for the comments
and suggestions that helped to improve the work.

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