# A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued
Lattices

Xiang-Gen Xia  X.-G. Xia is with the Department of Electrical and Computer
Engineering, University of Delaware, Newark, DE 19716, USA (e-mail:
xxia@ece.udel.edu).

###### Abstract

A matrix-valued inner product was proposed before to construct orthonormal
matrix-valued wavelets for matrix-valued signals. It introduces a weaker
orthogonality for matrix-valued signals than the orthogonality of all
components in a matrix that is commonly used in orthogonal multiwavelet
constructions. With the weaker orthogonality, it is easier to construct
orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-
valued inner product more from the inner product viewpoint that is more
fundamental and propose a new but equivalent norm for matrix-valued signals.
We show that although it is not scalar-valued, it maintains most of the
scalar-valued inner product properties. We introduce a new linear independence
concept for matrix-valued signals and present some related properties. We then
present the Gram-Schmidt orthonormalization procedure for a set of linearly
independent matrix-valued signals. Finally we define matrix-valued lattices,
where the newly introduced Gram-Schmidt orthogonalization might be applied.

###### Index Terms:

Matrix-valued inner product, matrix-valued signal space, nondegenerate matrix-
valued signals, linearly independent matrix-valued signals, matrix-valued
lattices, matrix-valued wavelets

##  I Introduction

Matrix-valued (vector-valued) signals are everywhere these days, such as,
videos, multi-spectral images, signals from multiarray multisensors, and high
dimensional data. For these signals, there are correlations not only over the
time but also across their matrix components. How to efficiently represent and
process them plays an important and fundamental role in data science.

In [1, 2], a matrix-valued inner product was introduced for matrix-valued (or
vector-valued) signals. It leads to a weaker orthogonality (called
Orthogonality B) than the component-wise orthogonality (called Orthogonality
A) in orthogonal multiwavelet constructions [11, 12]. The weaker
orthogonality, i.e., Orthogonality B, provides an easier sufficient condition
to construct orthonormal multiwavelets with Orthogonality B than the necessary
and sufficient condition [12] to construct orthonormal multiwavelets with
Orthogonality A. A connection between multiwavelets and matrix-valued/vector-
valued wavelets can be found in [2]. After the works in [1, 2], there have
been many studies on matrix-valued/vector-valued wavelets for matrix-
valued/vector-valued signals in the literature, see, for example, [3]-[10].

On the other hand, the Orthogonality B induced from the matrix-valued inner
product is stronger than the orthogonality, which is called Orthogonality C
here, induced from the commonly used scalar-valued inner product for matrices.
It is because, as we shall see later, Orthogonality B is basically the
orthogonality between all row vectors of two matrix-valued signals, while
Orthogonality C is the orthogonality of two long vectors of concatenated row
vectors of two matrix-valued signals. It was proved in [2] that Orthogonality
B or the matrix-valued inner product is able to completely decorrelate matrix-
valued signals not only in time domain but also across the components inside
matrix in the sense that it provides a complete Karhunen-Loève expansion for
matrix-valued signals, while Orthogonality A or Orthogonality C may not do so.
In other words, Orthogonality B induced from the matrix-valued inner product
is the proper orthogonality for matrix-valued signals. This also means that
the matrix-valued inner product is needed to study the decorrelation of
matrix-valued signals and a conventional scalar-valued inner product may not
be enough.

Since the main goal in [1, 2] was to construct orthonormal matrix-valued
(vector-valued) wavelets, not much about the inner product or the
orthogonality itself, which is more fundamental, was studied. In this paper,
we study more properties on the matrix-valued inner product and its induced
Orthogonality B for matrix-valued signal space proposed in [1, 2]. We first
define a different norm for matrix-valued signals than that defined in [2] and
prove that these two norms are equivalent. The norm defined in this paper is
consistent with the matrix-valued inner product similar to that for a scalar-
valued inner product. We introduce a new linear independence concept for
matrix-valued signals and present some related properties. We then present the
Gram-Schmidt orthonormalization procedure for a set of linearly independent
matrix-valued signals. We finally define matrix-valued lattices, where the
newly introduced Gram-Schmidt orthogonalization might be applied. Due to the
noncommuntativity of matrix multiplications, these concepts and properties for
matrix-valued signals and/or inner product are not straightforward extensions
of the conventional ones for scalar-valued signals and/or inner product.

The remainder of this paper is organized as follows. In Section II, we
introduce matrix-valued signal space, matrix-valued inner product, and define
a new norm for matrix-valued signals. We present some simple properties for
the matrix-valued inner product and prove that the new norm proposed in this
paper is equivalent to that used in [2]. In Section III, we first introduce
the concepts of degenerate and nondegenerate matrix-valued signals and then
introduce the concept of linear independence for matrix-valued signals. The
newly introduced linear independence is different from but consistent with the
conventional one for vectors. We also present some interesting properties on
the linear independence and the orthogonality. We finally present the Gram-
Schmidt orthonormalization procedure for a set of linearly independent matrix-
valued signals, which has the similar form as the conventional one for vectors
but not a straightforward generalization due to the noncommutativity of matrix
multiplications and the matrix-valued inner product used in the procedure. In
Section IV, we define matrix-valued lattices. In Section V, we conclude this
paper.

##  II Matrix-Valued Signal Space and Matrix-Valued Inner Product

We first introduce matrix-valued signal space studied in [1, 2]. Let
${\mathbb{C}}^{N\times N}$ denote all $N\times N$ matrices of complex-valued
entries, and for $-\infty\leq a<b\leq\infty$, let $L^{2}(a,b)$ denote all the
finite energy signals in the interval $(a,b)$ and

$L^{2}(a,b;{\mathbb{C}}^{N\times
N})\stackrel{{\scriptstyle\Delta}}{{=}}\left\\{{\bf
f}(t)=\left(\begin{array}[]{cccc}f_{11}(t)&f_{12}(t)&\cdots&f_{1N}(t)\\\
f_{21}(t)&f_{22}(t)&\cdots&f_{2N}(t)\\\ &&\cdots&\\\
f_{N1}(t)&f_{N2}(t)&\cdots&f_{NN}(t)\end{array}\right):\;\;f_{kl}\in
L^{2}(a,b),1\leq k,l\leq N\right\\}.$ (1)

We call $L^{2}(a,b;{\mathbb{C}}^{N\times N})$ a matrix-valued signal space and
$\mathbf{f}(t)\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$, or simply
$\mathbf{f}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$, a matrix-valued signal.

For any $A\in\mathbb{C}^{N\times N}$ and ${\bf f}\in
L^{2}(a,b;\mathbb{C}^{N\times N})$, the products $A{\bf f},\,{\bf f}A\in
L^{2}(a,b;\mathbb{C}^{N\times N})$. This implies that the matrix-valued signal
space $L^{2}(a,b;\mathbb{C}^{N\times N})$ is defined over $\mathbb{C}^{N\times
N}$ and not simply over $\mathbb{C}$. For $\mathbf{f}\in
L^{2}(a,b;{\mathbb{C}}^{N\times N})$, its integration
$\int_{a}^{b}\mathbf{f}(t)dt$ is defined by the integrations of its
components, i.e.,
$\int_{a}^{b}\mathbf{f}(t)dt=\left(\int_{a}^{b}f_{kl}(t)dt\right)$.

Let $\|\cdot\|_{M}$ denote a matrix norm on $\mathbb{C}^{N\times N}$, for
example, the Frobenius norm,

$\|A\|_{M}=\|A\|_{F}=\left(\sum_{k,l=1}^{N}|A_{kl}|^{2}\right)^{1/2},$

where $A=(A_{kl})$. For each ${\bf f}\in L^{2}(a,b;\mathbb{C}^{N\times N})$,
let $\|{\bf f}\|_{M}$ denote the norm of ${\bf f}$ associated with the matrix
norm $\|\cdot\|_{M}$ as

$\|{\bf
f}\|_{M}\stackrel{{\scriptstyle\Delta}}{{=}}\left\lVert\int_{a}^{b}{\bf
f}(t){\bf f}^{\dagger}(t)dt\right\rVert_{M}^{1/2},$ (2)

where † denotes the complex conjugate transpose. Note that the norm $\|{\bf
f}\|$ of ${\bf f}$ defined in [2] has the following form

$\|{\bf f}\|\stackrel{{\scriptstyle\Delta}}{{=}}\left(\int_{a}^{b}\|{\bf
f}(t)\|_{M}^{2}dt\right)^{1/2},$ (3)

where $\|{\bf f}(t)\|_{M}$ is the matrix norm of matrix ${\bf f}(t)$ for a
fixed $t$. We will show later that the above two norms $\|{\bf f}\|$ and
$\|{\bf f}\|_{M}$ are equivalent in the sense that there exist two positive
constants $C_{1}>0$ and $C_{2}>0$ such that

$C_{1}\|{\bf f}\|\leq\|{\bf f}\|_{M}\leq C_{2}\|{\bf f}\|,\mbox{ for any }{\bf
f}\in L^{2}(a,b;{\mathbb{C}}^{N\times N}).$ (4)

We next define matrix-valued inner product for matrix-valued signals in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$. For two matrix-valued signals ${\bf
f},{\bf g}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$, their matrix-valued inner
product (or simply inner product) $\langle{\bf f},{\bf g}\rangle$ is defined
as the integration of the matrix product ${\bf f}(t){\bf g}^{\dagger}(t)$,
i.e.,

$\langle{\bf f},{\bf
g}\rangle\stackrel{{\scriptstyle\Delta}}{{=}}\int_{a}^{b}{\bf f}(t){\bf
g}^{\dagger}(t)dt.$ (5)

With the definition (5), most properties of the conventional scalar-valued
inner product hold for the above matrix-valued inner product. For instance,
the following properties of the matrix-valued inner product are clear:

  * (i)

$\langle{\bf f},{\bf g}\rangle=\langle{\bf g},{\bf f}\rangle^{\dagger}$.

  * (ii)

$\langle{\bf f},{\bf f}\rangle=0$ if and only if ${\bf f}=0$.

  * (iii)

$\|{\bf f}\|_{M}=\|\langle{\bf f},{\bf f}\rangle\|_{M}^{1/2}$.

  * (iv)

For any $A,B\in{\mathbb{C}}^{N\times N}$, $\langle A{\bf f},B{\bf
g}\rangle=A\langle{\bf f},{\bf g}\rangle B^{\dagger}$.

Note that Property (iii) may not hold for the norm (3) used in [2].

Two matrix-valued signals ${\bf f}$ and ${\bf g}$ in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$ are called orthogonal if $\langle{\bf
f},{\bf g}\rangle=0$. A set of matrix-valued signals is called an orthogonal
set if any two distinct matrix-valued signals in the set are orthogonal. A
sequence $\mathbf{\Phi}_{k}(t)\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$,
$k\in{\mathbb{Z}}$, is called an orthonormal set in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$ if

$\langle\mathbf{\Phi}_{k},\mathbf{\Phi}_{l}\rangle=\delta(k-l)I_{N},\,\,\,k,l\in{\mathbb{Z}},$
(6)

where $\delta(k)=1$ when $k=0$ and $\delta(k)=0$ when $k\neq 0$, $I_{N}$ is
the $N\times N$ identity matrix. Due to (i) above, the
orthogonality/orthonormality between ${\bf f}$ and ${\bf g}$ is commutative,
i.e., if ${\bf f}$ and ${\bf g}$ are orthogonal/orthonormal, then ${\bf g}$
and ${\bf f}$ are orthogonal/orthonormal too.

A sequence $\mathbf{\Phi}_{k}(t)\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$,
$k\in{\mathbb{Z}}$, is called an orthonormal basis for
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$ if it satisfies (6), and moreover, for
any $\mathbf{f}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$ there exists a
sequence of $N\times N$ constant matrices $F_{k}$, $k\in\mathbb{Z}$, such that

$\mathbf{f}(t)=\sum_{k\in\mathbb{Z}}F_{k}\mathbf{\Phi}_{k}(t),\,\,\mbox{ for
}t\in[a,b],$ (7)

or simply

$\mathbf{f}=\sum_{k\in\mathbb{Z}}F_{k}\mathbf{\Phi}_{k},$

where $F_{k}=\langle\mathbf{f},\mathbf{\Phi}_{k}\rangle$, the multiplication
$F_{k}\mathbf{\Phi}_{k}(t)$ for each fixed $t$ is the $N\times N$ matrix
multiplication, and the convergence for the infinite summation is in the sense
of the norm $\|\cdot\|_{M}$ defined by (2) for the matrix-valued signal space.
The corresponding Parseval equality is

$\langle\mathbf{f},\mathbf{f}\rangle=\sum_{k\in\mathbb{Z}}F_{k}F^{\dagger}_{k}.$
(8)

With the norm $\|\cdot\|_{M}$ in (2), it is clear that for any element
$\mathbf{\Phi}_{k}$ in an orthonormal set in $L^{2}(a,b;{\mathbb{C}}^{N\times
N})$, we have $\|\mathbf{\Phi}_{k}\|_{M}=1$, which is consistent with the
conventional relationship between vector norm and vector inner product.
However, this property may not hold for the norm $\|\cdot\|$ in (3) used in
[2]. We refer to [2] for the Karhunen-Loève expansion with an orthonormal
basis for random processes of matrix-valued signals.

We next show the equivalence (4) of the two norms $\|\cdot\|_{M}$ in (2) and
$\|\cdot\|$ in (3).

###### Proposition 1

The norms $\|\cdot\|_{M}$ in (2) and $\|\cdot\|$ in (3) are equivalent.

Proof: It is known that all matrix norms for constant matrices are equivalent.
Hence, we show (4) only for the Frobenius norm, i.e.,
$\|\cdot\|_{M}=\|\cdot\|_{F}$. In this case, the following is an elementary
proof.

$\displaystyle\|{\bf f}\|_{M}^{4}$ $\displaystyle=$
$\displaystyle\sum_{k,l=1}^{N}\left|\int_{a}^{b}\sum_{m=1}^{N}f_{km}(t)f_{lm}^{*}(t)dt\right|^{2}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{N}\left|\sum_{m=1}^{N}\int_{a}^{b}|f_{km}(t)|^{2}dt\right|^{2}+\sum_{k\neq
l=1}^{N}\left|\sum_{m=1}^{N}\int_{a}^{b}f_{km}(t)f_{lm}^{*}(t)dt\right|^{2}$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{N}\left|\sum_{m=1}^{N}\int_{a}^{b}|f_{km}(t)|^{2}dt\right|^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{N}\left(\int_{a}^{b}\sum_{k,l=1}^{N}|f_{kl}(t)|^{2}dt\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{N}\|{\bf f}\|^{4}.$

Thus, we have

$\frac{1}{N^{1/4}}\|{\bf f}\|\leq\|{\bf f}\|_{M}.$ (9)

On the other hand,

$\displaystyle\|{\bf f}\|_{M}^{4}$ $\displaystyle=$
$\displaystyle\sum_{k,l=1}^{N}\left|\int_{a}^{b}\sum_{m=1}^{N}f_{km}(t)f_{lm}^{*}(t)dt\right|^{2}$
$\displaystyle\leq$ $\displaystyle
N\sum_{k,l=1}^{N}\sum_{m=1}^{N}\left|\int_{a}^{b}f_{km}(t)f_{lm}^{*}(t)dt\right|^{2}$
$\displaystyle\leq$ $\displaystyle
N\sum_{k,l=1}^{N}\sum_{m=1}^{N}\int_{a}^{b}|f_{km}(t)|^{2}dt\int_{a}^{b}|f_{lm}(t)|^{2}dt$
$\displaystyle\leq$
$\displaystyle\frac{N}{2}\sum_{k,l=1}^{N}\sum_{m=1}^{N}\left(\left(\int_{a}^{b}|f_{km}(t)|^{2}dt\right)^{2}+\left(\int_{a}^{b}|f_{lm}(t)|^{2}dt\right)^{2}\right)$
$\displaystyle\leq$ $\displaystyle
N^{2}\sum_{k=1}^{N}\sum_{m=1}^{N}\left(\int_{a}^{b}|f_{km}(t)|^{2}dt\right)^{2}$
$\displaystyle\leq$ $\displaystyle
N^{2}\left(\int_{a}^{b}\sum_{k,m=1}^{N}|f_{km}(t)|^{2}dt\right)^{2}$
$\displaystyle=$ $\displaystyle N^{2}\|{\bf f}\|^{4}.$

This shows that

$\|{\bf f}\|_{M}\leq\sqrt{N}\|{\bf f}\|.$ (10)

Combining (9) and (10), the equivalence (4) with $C_{1}=N^{-1/4}$ and
$C_{2}=N^{1/2}$ between the norms $\|\cdot\|_{M}$ in (2) and $\|\cdot\|$ in
(3) is proved. q.e.d.

Due to the equivalence of the norm $\|\cdot\|_{M}$ proposed in this paper and
the norm $\|\cdot\|$ used in [2], all the results on orthonormal matrix-valued
wavelets obtained in [2] hold, when the norm $\|\cdot\|_{M}$ for matrix-valued
signals in this paper is used.

As a remark, the conventional inner product for two matrices $A$ and $B$ is
the scalar-valued inner product $tr(AB^{\dagger})$ where $tr$ stands for the
matrix trace. It is not hard to see that with this scalar-valued inner
product, the orthogonality between two matrix-valued signals, which is called
Orthogonality C, is the orthogonality of two long vectors of concatenated row
vectors of two matrix-valued signals. As mentioned in Introduction, and it is
also not hard to see from the above definition, the orthogonality (6) induced
from the matrix-valued inner product in this paper for two matrix-valued
signals is the orthogonality between any row vectors including the row vectors
inside a matrix of the two matrix-valued signals, which is named Orthogonality
B in [2]. Clearly Orthogonality B is stronger than Orthogonality C, while it
is weaker than the component-wise orthogonality called Orthogonality A in [2],
commonly used in multiwavelets [11, 12].

With Orthogonality A, a necessary and sufficient condition to construct
orthonormal multiwavelets was given in [12] that is not easy to check.
However, with Orthogonality B, an easy sufficient condition to construct
orthonormal multiwavelets was obtained in [2]. Furthermore, it was shown in
[2] that the matrix-valued inner product (5) and its induced Orthogonality B
provide a complete decorrelation of matrix-valued signals along time and
across matrix components in the sense that a complete Karhunen-Loève expansion
for matrix-valued signals can be obtained. This may not be possible for
Orthogonality A or Orthogonality C induced from a scalar-valued inner product
[13, 14]. In other words, the matrix-valued inner product (5) is fundamental
to study matrix-valued signals.

##  III Linear Independence and Gram-Schmidt Orthonormalization

Let us first introduce degenerate and linearly independent matrix-valued
signals, and study their properties.

###  III-A Degenerate and Linearly Independent Matrix-Valued Signals

A matrix-valued signal ${\bf f}$ in $L^{2}(a,b;{\mathbb{C}}^{N\times N})$ is
called degenerate signal if $\langle{\bf f},{\bf f}\rangle$ does not have full
rank, otherwise it is called nondegenerate signal. A sequence of matrix-valued
signals ${\bf f}_{k}$ in $L^{2}(a,b;{\mathbb{C}}^{N\times N})$, $k=1,2,...,K$,
are called linearly independent if the following condition holds: if

$\sum_{k=1}^{K}F_{k}{\bf f}_{k}\stackrel{{\scriptstyle\Delta}}{{=}}{\bf f}$
(11)

for constant matrices $F_{k}\in\mathbb{C}^{N\times N}$, $k=1,2,...,K$, is
degenerate, then the null space of matrix $F_{k}^{\dagger}$ includes the null
space of matrix $\langle{\bf f},{\bf f}\rangle$ for every $k$, $k=1,2,...,K$.
Clearly, the above linear independence returns to the conventional one when
all the above matrices including both $F_{k}$ and ${\bf f}_{k}$ are diagonal.
Furthermore, if ${\bf f}=0$ in (11), the above condition implies that all
$F_{k}=0$, $k=1,2,...,K$, since in this case, the null space of $\langle{\bf
f},{\bf f}\rangle$ is the whole space $\mathbb{C}^{N\times N}$. This concides
with the condition of the conventional linear independence.

###### Proposition 2

If matrix-valued signals ${\bf f}_{k}$, $k=1,2,...,K$, are linearly
independent, then, all signals ${\bf f}_{k}$, $k=1,2,...,K$, are
nondegenerate.

Proof: Without loss of generality, assume ${\bf f}_{1}$ is degenerate. Let
$F_{1}=I_{N}$ and $F_{k}=0$ for $k=2,3,...,K$. Then, we have that

$\sum_{k=1}^{K}F_{k}{\bf f}_{k}={\bf f}_{1}$

is degenerate, while the null space of $F_{1}^{\dagger}$ is $0$ only and does
not include the null space of $\langle{\bf f}_{1},{\bf f}_{1}\rangle$. In
other words, ${\bf f}_{k}$, $k=1,2,...,K$, are not linearly independent. This
contradicts the assumption in the proposition and therefore the proposition is
proved. q.e.d.

As one can see, the above concept of degenerate signal is similar to that of
$0$ in the conventional linear dependence or independence.

###### Proposition 3

Let $G_{k}\in\mathbb{C}^{N\times N}$, $k=1,2,...,K$, be $K$ constant matrices
and at least one of them have full rank. If matrix-valued signals ${\bf
f}_{k}$, $k=1,2,...,K$, are linearly independent, then
$\sum_{k=1}^{K}G_{k}{\bf f}_{k}$ is nondegenerate.

Proof: Without loss of generality, let us assume $G_{1}$ has full rank. If
$\sum_{k=1}^{K}G_{k}{\bf f}_{k}={\bf g}$ is degenerate, then by the linear
independence of ${\bf f}_{k}$, $k=1,2,...,K$, the null space of
$G_{1}^{\dagger}$ cannot only contain $0$, which contradicts the assumption
that $G_{1}$ has full rank. q.e.d.

It is clear to see that Proposition 2 is a special case of Proposition 3.

###### Proposition 4

If matrix-valued signals ${\bf f}_{k}$, $k=1,2,...,K$, are linearly
independent, then for any full rank constant matrices
$G_{k}\in\mathbb{C}^{N\times N}$, $k=1,2,...,K$, matrix-valued signals ${\bf
g}_{k}\stackrel{{\scriptstyle\Delta}}{{=}}G_{k}{\bf f}_{k}$, $k=1,2,...,K$,
are also linearly independent.

Proof: For any constant matrices $F_{k}\in\mathbb{C}^{N\times N}$,
$k=1,2,...,K$, if

$\sum_{k=1}^{K}F_{k}{\bf g}_{k}=\sum_{k=1}^{K}F_{k}G_{k}{\bf f}_{k}={\bf f}$

is degenerate, then for each $k$, $1\leq k\leq K$, the null space of matrix
$(F_{k}G_{k})^{\dagger}=G_{k}^{\dagger}F_{k}^{\dagger}$ includes the null
space of matrix $\langle{\bf f},{\bf f}\rangle$, since ${\bf f}_{k}$,
$k=1,2,...,K$, are linearly independent. Because all matrices $G_{k}$,
$k=1,2,...,K$, have full rank, for each $k$, $1\leq k\leq K$, the null spaces
of $F_{k}^{\dagger}$ and $G_{k}^{\dagger}F_{k}^{\dagger}$ are the same, thus,
the null space of $F_{k}^{\dagger}$ includes the null space of $\langle{\bf
f},{\bf f}\rangle$ as well. This proves the proposition. q.e.d.

Similar to the conventional linear dependence of vectors, we have the
following result for matrix-valued signals.

###### Proposition 5

For a matrix-valued signal ${\bf f}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$
and two constant matrices $A,B\in\mathbb{C}^{N\times N}$, matrix-valued
signals $A{\bf f}$ and $B{\bf f}$ are linearly dependent.

Proof: If $A{\bf f}$ and $B{\bf f}$ are linearly independent, then, from
Proposition 2 it is easy to see that matrices $A$ and $B$ all have full rank
and ${\bf f}$ is nondegenerate. Then, we have

$BA^{-1}A{\bf f}-B{\bf f}=0,$

which contradicts with the assumption of the linear independence of $A{\bf f}$
and $B{\bf f}$. This proves the proposition. q.e.d.

Although it is obvious for the conventional vectors, the result in Proposition
5 for matrix-valued signals may not be so, due to the matrix-valued
coefficient multiplications as it can be seen from the above proof. We next
consider more general linear combinations of linearly independent matrix-
valued signals.

For $1\leq p\leq K$, let $S_{1},...,S_{p}$ be a partition of the index set
$\\{1,2,...,K\\}$ and each $S_{i}$ has $K_{i}$ elements, where $S_{i_{1}}\cap
S_{i_{2}}=\emptyset$ for $1\leq i_{1}\neq i_{2}\leq p$,
$\cup_{i=1}^{p}S_{i}=\\{1,2,...,K\\}$, and $1\leq K_{1},...,K_{p}\leq K$ with
$K_{1}+K_{2}+\cdots+K_{p}=K$.

###### Proposition 6

For each $i$, $1\leq i\leq p$, let $G_{k_{i}}\in\mathbb{C}^{N\times N}$,
$k_{i}\in S_{i}$, be $K_{i}$ constant matrices and at least one of them have
full rank. If matrix-valued signals ${\bf f}_{k}$, $k=1,2,...,K$, are linearly
independent, then the following $p$ matrix-valued signals:

$\sum_{k_{i}\in S_{i}}G_{k_{i}}{\bf f}_{k_{i}},\,\,\mbox{ for }i=1,2,...,p,$

are linearly independent.

Proof: Let $F_{i}\in\mathbb{C}^{N\times N}$, $i=1,2,...,p$, be constant
matrices. Assume that

$\sum_{i=1}^{p}F_{i}\sum_{k_{i}\in S_{i}}G_{k_{i}}{\bf f}_{k_{i}}={\bf g}$

is degenerate. Then,

$\sum_{i=1}^{p}\sum_{k_{i}\in S_{i}}F_{i}G_{k_{i}}{\bf f}_{k_{i}}={\bf g},$

and by the linear independence of ${\bf f}_{k}$, $k=1,2,...,K$, we know that
the null space of $(F_{i}G_{k_{i}})^{\dagger}$ for every $k_{i}\in S_{i}$ and
every $i=1,...,p$ contains the null space of matrix $\langle{\bf g},{\bf
g}\rangle$. From the condition in the proposition, without loss of generality,
we may assume that $G_{k_{i,1}}$, for some $k_{i,1}\in S_{i}$, has full rank
for $1\leq i\leq p$. Thus, the null space of $(F_{i}G_{k_{i,1}})^{\dagger}$,
or $G_{k_{i,1}}^{\dagger}F_{i}^{\dagger}$, contains the null space of
$\langle{\bf g},{\bf g}\rangle$ for $1\leq i\leq p$. Since $G_{k_{i,1}}$ has
full rank, the null space of $F_{i}^{\dagger}$ must contain the null space of
$\langle{\bf g},{\bf g}\rangle$ for $1\leq i\leq p$. This proves the
proposition. q.e.d.

Note that when $p=1$ in Proposition 6, it returns to Proposition 3, and when
$p=K$ in Proposition 6, it returns to Proposition 4.

###### Proposition 7

If ${\bf f}_{k}$, $k=1,2,...,K$, form an orthonormal set in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$, then, they must be linearly
independent.

Proof: For constant matrices $F_{k}\in{\mathbb{C}}^{N\times N}$,
$k=1,2,...,K$, let

$\sum_{k=1}^{K}F_{k}{\bf f}_{k}={\bf f}.$

Then, from the Parseval equality (8), we have

$\sum_{k=1}^{K}F_{k}F_{k}^{\dagger}=\langle{\bf f},{\bf f}\rangle.$

Assume that for some vector $u\neq 0$, we have $\langle{\bf f},{\bf f}\rangle
u=0$ but $F_{k_{0}}^{\dagger}u\neq 0$ for some $k_{0}$, $1\leq k_{0}\leq K$.
Then,

$0<u^{\dagger}F_{k_{0}}F_{k_{0}}^{\dagger}u\leq\sum_{k=1}^{K}u^{\dagger}F_{k}F_{k}^{\dagger}u=u^{\dagger}\langle{\bf
f},{\bf f}\rangle u=0,$

which leads to a contradiction. Thus, for each $k$, $1\leq k\leq K$, the null
space of $F_{k}^{\dagger}$ includes the null space of $\langle{\bf f},{\bf
f}\rangle$. This proves the linear independence of ${\bf f}_{k}$,
$k=1,2,...,K$. q.e.d.

###### Corollary 1

Assume ${\bf f}_{k}$, $k=1,2,...,K$, are nondegenerate matrix-valued signals
and form an orthogonal set in $L^{2}(a,b;{\mathbb{C}}^{N\times N})$. Then,
${\bf g}_{k}\stackrel{{\scriptstyle\Delta}}{{=}}\langle{\bf f}_{k},{\bf
f}_{k}\rangle^{-1/2}{\bf f}_{k}$, $k=1,2,...,K$, form an orthonormal set in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$, and ${\bf f}_{k}$, $k=1,2,...,K$, are
linearly independent.

Proof: Since all matrix-valued signals ${\bf f}_{k}$ are nondegenerate,
matrices $\langle{\bf f}_{k},{\bf f}_{k}\rangle$ all have full rank. From the
property (iv) for the matrix-valued inner product and $\\{{\bf f}_{k}\\}$ is
an orthogonal set, for every $k,l$, $1\leq k,l\leq K$,

$\langle{\bf g}_{k},{\bf g}_{l}\rangle=\langle{\bf f}_{k},{\bf
f}_{k}\rangle^{-1/2}\langle{\bf f}_{k},{\bf f}_{l}\rangle\langle{\bf
f}_{l},{\bf f}_{l}\rangle^{-1/2}=\delta(k-l)I_{N}.$

Thus, ${\bf g}_{k}$, $k=1,2,...,K$, form an orthonormal set in
$L^{2}(a,b;{\mathbb{C}}^{N\times N})$.

Then, the linear independence of ${\bf f}_{k}=\langle{\bf f}_{k},{\bf
f}_{k}\rangle^{1/2}{\bf g}_{k}$, $k=1,2,...,K$, immediately follows from
Propositions 4 and 7. q.e.d.

The result in Corollary 1 is consistent with the conventional one for vectors,
i.e., any orthogonal set of nonzero vectors must be linearly independent.
However, there is a difference. In the above relationship between
orthogonality and linear independence, matrix-valued signals need to be
nondegenerate. Note that it is possible that a matrix-valued signal ${\bf f}$
in an orthogonal set in $L^{2}(a,b;{\mathbb{C}}^{N\times N})$ is degenerate,
i.e., $\langle{\bf f},{\bf f}\rangle$ may not necessarily have full rank, even
though ${\bf f}\neq 0$. Thus, a general orthogonal set of matrix-valued
signals may not have to be linearly independent. This does not occur for any
orthogonal set of nonzero signals when a scalar-valued inner product is used.

###  III-B Gram-Schmidt Orthonormalization

We are now ready to present the Gram-Schmidt orthonormalization for a finite
sequence of linearly independent matrix-valued signals. Let ${\bf f}_{k}\in
L^{2}(a,b;{\mathbb{C}}^{N\times N})$, $k=1,2,...,K$, be linearly independent.
The Gram-Schmidt orthonormalization for this sequence is as follows, which is
similar to, but not a straightforward extension of, the conventional one, due
to the noncommutativity of matrix multiplications.

Since ${\bf f}_{k}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$, $k=1,2,...,K$, are
linearly independent, by Proposition 2, ${\bf f}_{1}$ is nondegenerate, i.e.,
matrix $\langle{\bf f}_{1},{\bf f}_{1}\rangle$ is invertible and positive
definite. Let

${\bf g}_{1}=\langle{\bf f}_{1},{\bf f}_{1}\rangle^{-1/2}{\bf f}_{1}.$ (12)

Then, we have

$\displaystyle\langle{\bf g}_{1},{\bf g}_{1}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bf f}_{1},{\bf f}_{1}\rangle^{-1/2}\int_{a}^{b}{\bf
f}_{1}(t){\bf f}_{1}^{\dagger}(t)dt\langle{\bf f}_{1},{\bf
f}_{1}\rangle^{-1/2}$ (13) $\displaystyle=$ $\displaystyle\langle{\bf
f}_{1},{\bf f}_{1}\rangle^{-1/2}\langle{\bf f}_{1},{\bf
f}_{1}\rangle\langle{\bf f}_{1},{\bf f}_{1}\rangle^{-1/2}=I_{N}.$

Let

$\displaystyle\hat{\bf g}_{2}$ $\displaystyle=$ $\displaystyle{\bf
f}_{2}-\langle{\bf f_{2}},{\bf g}_{1}\rangle{\bf g}_{1},$ (14)
$\displaystyle{\bf g}_{2}$ $\displaystyle=$ $\displaystyle\langle\hat{\bf
g}_{2},\hat{\bf g}_{2}\rangle^{-1/2}\hat{\bf g}_{2}.$ (15)

For (15) to be vaild, we need to show that $\hat{\bf g}_{2}$ in (14) is
nondegenerate. In fact, if $\hat{\bf g}_{2}$ is degenerate, then, from (14)
and (12), we have

$\displaystyle\hat{\bf g}_{2}$ $\displaystyle=$ $\displaystyle{\bf
f}_{2}-\langle{\bf f}_{2},{\bf g}_{1}\rangle\langle{\bf f}_{1},{\bf
f}_{1}\rangle^{-1/2}{\bf f}_{1}$ $\displaystyle=$ $\displaystyle F_{1}{\bf
f}_{1}+F_{2}{\bf f}_{2},$

where $F_{2}=I_{N}$ and $F_{1}=-\langle{\bf f}_{2},{\bf
g}_{1}\rangle\langle{\bf f}_{1},{\bf f}_{1}\rangle^{-1/2}$. Similar to the
proof of Proposition 2, this contradicts the assumption that ${\bf f}_{1}$ and
${\bf f}_{2}$ are linearly independent. Therefore, it proves that $\hat{\bf
g}_{2}$ in (14) is nondegenerate and (15) is well-defined.

Let us then check the orthogonality between ${\bf g}_{1}$ and $\hat{\bf
g}_{2}$. From (14) and (13), we have

$\displaystyle\langle\hat{\bf g}_{2},{\bf g}_{1}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bf f}_{2},{\bf g}_{1}\rangle-\langle{\bf f}_{2},{\bf
g}_{1}\rangle\langle{\bf g}_{1},{\bf g}_{1}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bf f}_{2},{\bf g}_{1}\rangle-\langle{\bf f}_{2},{\bf
g}_{1}\rangle=0.$

From (15) and (13), we have that ${\bf g}_{1}$ and ${\bf g}_{2}$ form an
orthonormal set.

Repeat the above process and for a general $k$, $2\leq k\leq K$, we let

$\displaystyle\hat{\bf g}_{k}$ $\displaystyle=$ $\displaystyle{\bf
f}_{k}-\sum_{l=1}^{k-1}\langle{\bf f}_{k},{\bf g}_{l}\rangle{\bf g}_{l},$ (16)
$\displaystyle{\bf g}_{k}$ $\displaystyle=$ $\displaystyle\langle\hat{\bf
g}_{k},\hat{\bf g}_{k}\rangle^{-1/2}\hat{\bf g}_{k}.$ (17)

With the same proof as the above ${\bf g}_{1}$ and ${\bf g}_{2}$, we have the
following proposition.

###### Proposition 8

For a linearly independent set of matrix-valued signals ${\bf f}_{k}$,
$k=1,2,...,K$, let ${\bf g}_{1},{\bf g}_{2},...,{\bf g}_{K}$ be constructed in
(12) and (14)-(17). Then, ${\bf g}_{1},{\bf g}_{2},...,{\bf g}_{K}$ form an
orthonormal set.

As we can see, although the above Gram-Schmidt orthonormalization procedure
for matrix-valued signals is similar to the conventional one for vectors, it
is not a straightforward generalization due to 1) the noncommuntativity of
matrix mulitpications and 2) the matrix-valued inner product used in the above
procedure.

We also want to make a comment on the nondegenerate and linear independence
for matrix-valued signals. The condition for nondegenerate matrix-valued
signals is a weak condition. Unless the row vectors of functions are linearly
dependent in the conventional sense, otherwise, a matrix-valued signal is
usually nondegenerate. This can be seen from the following result.

###### Proposition 9

A matrix-valued signal ${\bf f}\in L^{2}(a,b;{\mathbb{C}}^{N\times N})$ is
degenerate if and only if its row vectors of functions in $L^{2}(a,b)$ are
linearly dependent.

Proof: A matrix-valued signal ${\bf f}$ is degenerate if and only if
$\langle{\bf f},{\bf f}\rangle$ does not have full rank. Since constant matrix
$\langle{\bf f},{\bf f}\rangle$ is Hermitian, it is true if and only if there
is a vector $u\neq 0\in\mathbb{C}^{N}$ such that $\langle u,\langle{\bf
f},{\bf f}\rangle u\rangle=0$, i.e.,

$\int_{a}^{b}\langle u,{\bf f}(t){\bf f}^{\dagger}(t)u\rangle
dt=\int_{a}^{b}\|{\bf f}^{\dagger}(t)u\|_{F}^{2}dt=0,$

which holds if and only if ${\bf f}^{\dagger}(t)u=0$ or $u^{\dagger}{\bf
f}(t)=0$ almost everywhere for $t\in(a,b)$, i.e., the row vectors of functions
of ${\bf f}(t)$ are linear dependent. q.e.d.

Thus, for a finite set of nondegenerate matrix-valued signals, they usually
satisfy the condition for linear independence for matrix-valued signals
defined above, i.e., they are usually linearly independent and therefore, they
can be made to an orthonormal set by using the above Gram-Schmidt procedure.

Another comment on the linear independence for matrix-valued signals is that
the definition in (11) is only for left multiplication of constant matrices
$F_{k}$ to matrix-valued signals ${\bf f}_{k}$. Similar definitions for linear
independences of matrix-valued signals with right constant matrix
multiplications and/or mixed left and right constant matrix multiplications
may be possible. Although what is studied in this paper is for continuous-time
matrix-valued signals, it can be easily generalized to discrete-time matrix-
valued signals (sequences of finite or infinite length).

##  IV Matrix-Valued Lattices

In this section, based on the matrix-valued signal space with the matrix-
valued inner product, we introduce matrix-valued lattices.

We first introduce matrix-valued lattices. For convenience, in what follows we
only consider the Frobenius norm for matrices, i.e.,
$\|\cdot\|_{M}=\|\cdot\|_{F}$, and real matrix-valued signal spaces, and also
let ${\cal R}$ denote the real matrix-valued signal space ${\cal
R}\stackrel{{\scriptstyle\Delta}}{{=}}L^{2}(a,b;{\mathbb{R}}^{N\times N})$.
Let $\mathbb{Z}^{N\times N}$ denote all $N\times N$ matrices of integer
entries.

For a finitely many linearly independent real matrix-valued signals ${\bf
f}_{k}\in{\cal R}$, $k=1,2,...,K$, let ${\cal R}^{K}$ denote the matrix-valued
signal space linearly expanded by them, i.e.,

${\cal R}^{K}=\left\\{\sum_{k=1}^{K}F_{k}{\bf
f}_{k}:\,\,F_{k}\in\mathbb{R}^{N\times N},\,k=1,2,...,K\right\\}.$ (18)

From what was studied in the previous section, clearly, ${\bf f}_{k}$,
$k=1,2,...,K$, form a basis in ${\cal R}^{K}$. The matrix-valued lattice
formed by this basis in ${\cal R}^{K}$ is defined as

${\cal L}=\left\\{\sum_{k=1}^{K}F_{k}{\bf
f}_{k}:\,\,F_{k}\in\mathbb{Z}^{N\times N},\,k=1,2,...,K\right\\},$ (19)

which is a subset/subgroup of ${\cal R}^{K}$. The basis ${\bf f}_{k}$,
$k=1,2,...,K$, is called a basis for the $K$ dimensional matrix-valued lattice
${\cal L}$.

The fundamental region of this lattice ${\cal L}$ can be defined similar to
the conventional lattice as follows. A set ${\cal F}\subset{\cal R}^{K}$ is
called a fundamental region, if its translations ${\bf x}+{\cal F}=\\{{\bf
x}+{\bf f}:\,\,{\bf f}\in{\cal F}\\}$ for ${\bf x}\in{\cal L}$ form a
partition of ${\cal R}^{K}$. Since the basis elements ${\bf f}_{k}$,
$k=1,2,...,K$, are not constant real vectors as in the conventional lattices,
it would not be convenient to define the determinant of the lattice. However,
with the Gram-Schmidt orthonormalization developed in the previous section, we
may define the determinant of the lattice directly as

$\det({\cal L})=\prod_{k=1}^{K}\|\hat{\bf f}_{k}\|_{F},$ (20)

where $\hat{\bf f}_{k}$, $k=1,2,...,K$, are from the following Gram-Schmidt
orthogonalization of ${\bf f}_{k}$, $k=1,2,...,K$, which is from the Gram-
Schmidt orthonormalization in the previous section:

$\hat{\bf f}_{1}={\bf f}_{1},$ $\hat{\bf f}_{k}=\hat{\bf g}_{k}={\bf
f}_{k}-\sum_{l=1}^{k-1}\mu_{l,k}\hat{\bf f}_{l},$ (21)

where

$\mu_{l,k}=\langle{\bf f}_{k},\hat{\bf f}_{l}\rangle\langle\hat{\bf
f}_{l},\hat{\bf f}_{l}\rangle^{-1},\,\,l=1,2,...,k-1\mbox{ and }k=2,3,...,K.$
(22)

It is clear to see that the spaces linearly spanned by $\\{{\bf f}_{1},{\bf
f}_{2},...,{\bf f}_{K}\\}$ and $\\{\hat{\bf f}_{1},\hat{\bf
f}_{2},...,\hat{\bf f}_{K}\\}$ are the same, i.e., ${\cal R}^{K}$ in (18),
since they can be linearly (over $\mathbb{R}^{N\times N}$) represented by each
other similar to the conventional vectors.

From the Gram-Schmidt orthogonalization (21), we have

$\langle{\bf f}_{k},{\bf f}_{k}\rangle=\langle\hat{\bf f}_{k},\hat{\bf
f}_{k}\rangle+\sum_{l=1}^{k-1}\mu_{l,k}\langle\hat{\bf f}_{l},\hat{\bf
f}_{l}\rangle\mu_{l,k}^{\dagger},$ (23)

for $k=1,2,...,K$. Using Property (iii) in Section II, the identity (23)
implies

$\|{\bf f}_{k}\|_{F}^{2}\leq\|\hat{\bf
f}_{k}\|_{F}^{2}+\sum_{l=1}^{k-1}\|\mu_{l,k}\|_{F}^{2}\cdot\|\hat{\bf
f}_{l}\|_{F}^{2},$ (24)

for $k=1,2,...,K$. From (23), it is also clear that $\|{\bf
f}_{k}\|_{F}\geq\|\hat{\bf f}_{k}\|_{F}$, $k=2,...,K$.

As a remark, we know that the conventional Gram-Schmidt orthogonalization
plays an important role in the LLL algorithm for the conventional lattice
basis reduction [20]. It is, however, not clear how the Gram-Schmidt
orthogonalization for matrix-valued signals introduced above can be applied in
matrix-valued lattice basis reduction.

##  V Conclusion

In this paper, we re-studied the matrix-valued inner product previously
proposed to construct orthonormal matrix-valued wavelets for matrix-valued
signal analysis [1, 2] where not much on the matrix-valued inner product or
its induced Orthogonality B, which is more fundamental, was studied. In order
to study more on the matrix-valued inner product and its induced Orthogonality
B, we first proposed a new norm for matrix-valued signals, which is more
consistent with the matrix-valued inner product than that used in [2], and is
similar to that with the conventional scalar-valued inner product. We showed
that these two norms are equivalent, which means that with the newly proposed
norm, all the results for contructing orthonormal matrix-valued wavelets
obtained in [2] still hold. We then proposed the concepts of degenerate and
nondegenerate matrix-valued signals and defined the linear independence for
matrix-valued signals, which is different from but similar to the conventional
linear independence for vectors. We also presented some properties on the
linear independence and the orthogonality. We then presented the Gram-Schmidt
orthonormalization procedure for a set of linearly independent matrix-valued
signals. Although this procedure is similar to the conventional one for
vectors, due to the noncommutativity of matrix multiplications and the matrix-
valued inner product used in the procedure, it is not a straightforward
generalization. We finally defined matrix-valued lattices, where the newly
introduced Gram-Schmidt orthogonalization might be applied.

Since it was shown in [2] that the matrix-valued inner product and
Orthogonality B provide a complete Karhunen-Loève expansion for matrix-valued
signals, which a scalar-valued inner product may not do, it is believed that
what was studied in this paper for matrix-valued inner product for matrix-
valued signal space will have fundamental applications for high dimensional
signal analysis in data science.

As a final note, after this paper was written, it has been found that the
matrix-valued signal space with the matrix-valued inner product in this paper
is related to Hilbert modules, see, for example, [15]-[19]. Interestingly, it
was mentioned in [18] that there does not exist any general notion of
“$C^{*}$-linear independence” due to the existence of zero-divisors. We
believe that the linear independence for matrix-valued signals introduced in
this paper is novel.

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# Inertia of Kwong matrices

Rajendra Bhatia  Ashoka University, Sonepat
Haryana, 131029, India<EMAIL_ADDRESS>and  Tanvi Jain  Indian
Statistical Institute, New Delhi 110016, India<EMAIL_ADDRESS>
###### Abstract.

Let $r$ be any real number and for any $n$ let $p_{1},\ldots,p_{n}$ be
distinct positive numbers. A Kwong matrix is the $n\times n$ matrix whose
$(i,j)$ entry is $(p_{i}^{r}+p_{j}^{r})/(p_{i}+p_{j}).$ We determine the
signatures of eigenvalues of all such matrices. The corresponding problem for
the family of Loewner matrices
$\begin{bmatrix}(p_{i}^{r}-p_{j}^{r})/(p_{i}-p_{j})\end{bmatrix}$ has been
solved earlier.

###### Key words and phrases:

Kwong Matrix, inertia, positive definite matrix, conditionally positive
definite matrix, Loewner matrix, Sylvester’s law, Vandermonde matrix.

###### 2010 Mathematics Subject Classification:

15A18, 15A48, 15A57, 42A82

##  1\. Introduction

Let $f$ be a nonnegative $C^{1}$ function on $(0,\infty).$ Let $n$ be a
positive integer and $p_{1}<p_{2}<\cdots<p_{n}$ distinct positive real
numbers. The $n\times n$ matrix

$L_{f}(p_{1},\ldots,p_{n})=\begin{bmatrix}\frac{f(p_{i})-f(p_{j})}{p_{i}-p_{j}}\end{bmatrix}$
(1)

is called a Loewner matrix associated with $f.$ These matrices play an
important role in several areas of analysis, one of them being Loewner’s
theory of operator monotone functions. A central theorem in this theory
asserts that $f$ is operator monotone if and only if all Loewner matrices
associated with $f$ are positive semidefinite. See [2], [3] and [14].

Closely related to Loewner matrices are the matrices

$K_{f}(p_{1},\ldots,p_{n})=\begin{bmatrix}\frac{f(p_{i})+f(p_{j})}{p_{i}+p_{j}}\end{bmatrix}.$
(2)

These too have been studied in several papers. In [10] Kwong showed that all
matrices $K_{f}$ are positive semidefinite if (but not only if) $f$ is
operator monotone. Because of this the matrices $K_{f}$ are sometimes called
Kwong matrices. Audenaert [1] has characterised all functions $f$ for which
all $K_{f}$ are positive semidefinite.

Of particular interest are the functions $f(t)=t^{r},$ where $r$ is any real
number. For these functions we denote $L_{f}$ and $K_{f}$ by $L_{r}$ and
$K_{r},$ respectively. Thus

$L_{r}(p_{1},\ldots,p_{n})=\begin{bmatrix}\frac{p_{i}^{r}-p_{j}^{r}}{p_{i}-p_{j}}\end{bmatrix},$
(3)

and

$K_{r}(p_{1},\ldots,p_{n})=\begin{bmatrix}\frac{p_{i}^{r}+p_{j}^{r}}{p_{i}+p_{j}}\end{bmatrix}.$
(4)

Another fundamental theorem of Loewner’s says that the function $f(t)=t^{r}$
is operator monotone if and only if $0\leq r\leq 1.$ Thus all matrices $L_{r}$
are positive semidefinite if and only if $0\leq r\leq 1.$ From the work of
Kwong and Audenaert cited above it follows that all matrices $K_{r}$ are
positive semidefinite if and only if $-1\leq r\leq 1$ and positive definite if
and only if $-1<r<1.$

In their work[5] Bhatia and Holbrook studied the matrices $L_{r}$ for values
of $r$ outside the interval $[0,1].$ Among other things, they showed that when
$1<r<2$ every matrix $L_{r}$ has exactly one positive eigenvalue. This is in
striking contrast to the case $0<r<1,$ in which all eigenvalues of $L_{r}$ are
positive. This led the authors [5] to make a conjecture about the signature of
eigenvalues of $L_{r}$ as $r$ varies over real numbers.

Let $A$ be any $n\times n$ Hermitian matrix. The inertia of $A$ is the triple

${\rm In}(A)=(\pi(A),\zeta(A),\nu(A)),$

where $\pi(A),$ $\zeta(A)$ and $\nu(A)$ are respectively the numbers of
positive, zero and negative eigenvalues of $A.$ By the results of Loewner
cited above $\text{\rm In}\,L_{r}=(n,0,0)$ when $0<r<1,$ and the result of
Bhatia-Holbrook says that $\text{\rm In}\,L_{r}=(1,0,n-1)$ when $1<r<2.$ The
conjecture in [5] described the inertia of $L_{r}$ for other values of $r.$

In [7] Bhatia and Sano made two essential contributions to this problem. They
provided a better understanding of the problem for the range $1<r<2,$ and they
also obtained a solution for the range $2<r<3.$ Let $\mathcal{H}_{1}$ be the
$(n-1)$-dimensional subspace of $\mathbb{C}^{n}$ defined as

$\mathcal{H}_{1}=\left\\{x\in\mathbb{C}^{n}:\sum_{i=1}^{n}x_{i}=0\right\\}.$
(5)

An $n\times n$ Hermitian matrix is said to be conditionally positive definite
(cpd) if $\langle x,Ax\rangle\geq 0$ for all $x\in\mathcal{H}_{1}.$ It is said
to be conditionally negative definite (cnd) if $-A$ is cpd. If $A$ is
nonsingular and cnd with all entries nonnegative, then $\text{\rm
In}\,A=(1,0,n-1).$ Bhatia and Sano [7] showed that the matrix $L_{r}$ is cnd
when $1<r<2,$ thus explaining the result in [5]. They also showed that $L_{r}$
is cpd when $2<r<3.$

In the same paper [7] the authors found an interesting difference between the
inertial properties of $L_{r}$ and $K_{r}$ in the range $2<r<3.$ They showed
that $K_{r}$ is nonsingular and cnd in the interval $1<r<3,$ and hence
$\text{\rm In}\,K_{r}=(1,0,n-1)$ for such $r.$ Thus, there arises the problem
of studying $\text{\rm In}\,K_{r}$ parallel to that of $\text{\rm In}\,L_{r}.$

The inertia of $L_{r}$ was completely determined by Bhatia, Friedland and Jain
in [4]. The corresponding theorem on $K_{r}$ was proved by us shortly
afterwards. This was announced in [6]. The aim of the present paper is to
publish our proof. Our main result is the following:

######  Theorem 1.

Let $p_{1}<p_{2}<\cdots<p_{n}$ and $r$ be any positive real numbers and let
$K_{r}$ be the matrix defined in (4). Then

  * (i)

$K_{r}$ is singular if and only if $r$ is an odd integer smaller than $n.$

  * (ii)

When $r$ is an odd integer smaller than or equal to $n,$ the inertia of
$K_{r}$ is given as follows

$\text{\rm
In}\,K_{r}=\begin{cases}\left(\lceil\frac{r}{2}\rceil,n-r,\lfloor\frac{r}{2}\rfloor\right)&r=1(\,\textrm{mod}\,4)\\\
\left(\lfloor\frac{r}{2}\rfloor,n-r,\lceil\frac{r}{2}\rceil\right)&r=3\,(\textrm{mod}\,4).\end{cases}$

  * (iii)

Suppose $k<r<k+2<n,$ where $k$ is an odd integer. Then

$\text{\rm
In}\,K_{r}=\begin{cases}\left(\lceil\frac{k}{2}\rceil,0,n-\lceil\frac{k}{2}\rceil\right)&k=1\,(\textrm{mod}\,4)\\\
\left(n-\lceil\frac{k}{2}\rceil,0,\lceil\frac{k}{2}\rceil\right)&k=3\,(\textrm{mod}\,4).\end{cases}$

  * (iv)

If $n$ is odd, then $\text{\rm In}\,K_{r}=\text{\rm In}\,K_{n}$ for $r>n-2$;
and if $n$ is even, then $\text{\rm In}\,K_{r}=\text{\rm
In}\,K_{n}=\left(\frac{n}{2},0,\frac{n}{2}\right)$ for $r>n-1.$

There is a striking similarity and a striking difference between the behaviour
of the signs of eigenvalues of $L_{r}$ and $K_{r}.$ As $r$ moves over
$(0,\infty)$ the eigenvalues of both flip signs at certain integral values of
$r.$ For $L_{r}$ these flips take place at all integers $r\leq n-1,$ and each
time all but one eigenvalue change signs. For $K_{r}$ the flips take place at
all odd integers $r\leq n-1.$ At $r=1$ all but one eigenvalue change signs,
and after that all but two eigenvalues change signs.

Figure 1.

Figure 1 is a schematic representation of the eigenvalues of $K_{r}$ for $n=6$
and $r\geq 0.$ In an earlier paper[6] we studied the inertia of another family
$P_{r}=\begin{bmatrix}(p_{i}+p_{j})^{r}\end{bmatrix}.$ The structure of the
proof there has been the template of subsequent works on this kind of problem.
See, e.g., [4] and the recent work [13] on the Kraus matrix. (Warning$:$ The
authors [13] use the symbol $K_{r}$ for something different from our Kwong
matrix). Our proof here follows the same steps as in these papers; the details
are different at some crucial points.

##  2\. Proof of Theorem 1

Two Hermitian matrices $A$ and $B$ are said to be congruent if there exists an
invertible matrix $X$ such that $B=X^{*}AX.$ The Sylvester law of inertia says
that $A$ and $B$ are congruent if and only if $\text{\rm In}\,A=\text{\rm
In}\,B.$ Let $D$ be the diagonal matrix $D=\text{\rm
diag}(p_{1},\ldots,p_{n}).$ Then for every $r>0$

$K_{-r}=D^{-r}K_{r}D^{-r},$ (6)

and hence,

$\text{\rm In}\,K_{-r}=\text{\rm In}\,K_{r}.$ (7)

The substitution $p_{i}=\text{\rm e}^{2x_{i}},$ $x_{i}\in\mathbb{R},$ gives

$K_{r}=\Delta\tilde{K}_{r}\Delta,$ (8)

where $\Delta=\text{\rm diag}\left(\text{\rm e}^{(r-1)x_{1}},\ldots,\text{\rm
e}^{(r-1)x_{n}}\right),$ and

$\tilde{K}_{r}=\begin{bmatrix}\frac{\cosh\,r(x_{i}-x_{j})}{\cosh\,(x_{i}-x_{j})}\end{bmatrix}.$
(9)

By Sylvester’s law $\text{\rm In}\,\tilde{K}_{r}=\text{\rm In}\,K_{r}.$ When
$n=2$ we have

$\tilde{K}_{r}=\begin{bmatrix}1&\frac{\cosh\,r(x_{1}-x_{2})}{\cosh\,(x_{1}-x_{2})}\\\
\frac{\cosh\,r(x_{1}-x_{2})}{\cosh\,(x_{1}-x_{2})}&1\end{bmatrix}.$

So,
$\det\,\tilde{K}_{r}=1-\frac{\cosh^{2}r(x_{1}-x_{2})}{\cosh^{2}(x_{1}-x_{2})}.$
This is positive if $0<r<1,$ zero if $r=1,$ and negative if $r>1.$ The inertia
of $K_{r}$ is $(2,0,0)$ in the first case, $(1,1,0)$ in the second case, and
$(1,0,1)$ in the third. All assertions of Theorem 1 are thus valid in the case
$n=2.$ We will use the following extension of Sylvester’s law. A proof is
given in [6].

######  Proposition 2.

Let $n\geq r,$ and let $A$ be an $r\times r$ Hermitian matrix and $X$ an
$r\times n$ matrix of rank $r.$ Then

${\rm In}\,\,X^{\ast}AX={\rm In}\,\,A+(0,n-r,0).$ (10)

We now prove part (ii) of the theorem. Let $r$ be an odd integer, $r\leq n.$
Then

$\frac{p_{i}^{r}+p_{j}^{r}}{p_{i}+p_{j}}=p_{i}^{r-1}-p_{i}^{r-2}p_{j}+p_{i}^{r-3}p_{j}^{2}-\cdots+p_{j}^{r-1}.$

So the matrix $K_{r}$ can be factored as

$K_{r}=W^{*}VW,$

where $W$ is the $r\times n$ Vandermonde matrix given by

$W=\begin{bmatrix}1&1&\cdots&1\\\ p_{1}&p_{2}&\cdots&p_{n}\\\
\vdots&\vdots&\vdots\vdots\vdots&\vdots\\\
p_{1}^{r-1}&p_{2}^{r-1}&\cdots&p_{n}^{r-1}\end{bmatrix}.$

and $V$ is the $r\times r$ antidiagonal matrix with entries
$(1,-1,1,-1,\cdots,-1,1)$ down its sinister diagonal. So by the generalised
Sylvester’s law (10) we have for every odd integer $r\leq n,$

$\text{\rm In}\,K_{r}=\text{\rm In}\,V+(0,n-r,0).$

The matrix $V$ is nonsingular and its eigenvalues are $\pm 1.$ In the case
$r=1\,(\textrm{mod}\,4),$ $\text{\rm tr}\,V=1$ and the multiplicity of $1$ as
an eigenvalue of $V$ exceeds by one the multiplicity of $-1.$ In the case
$r=3\,(\textrm{mod}\,4),$ $\text{\rm tr}\,V=-1$ and the multiplicity of $-1$
as an eigenvalue of $V$ exceeds by one the multiplicity of $1.$ This
establishes part (ii) of Theorem 1. Next, let $c_{1},c_{2},\ldots,c_{n}$ be
real numbers, not all of which are zero, and let $f$ be the function on
$(0,\infty)$ defined as

$f(x)=\sum_{j=1}^{n}c_{j}\frac{x^{r}+p_{j}^{r}}{x+p_{j}}.$ (11)

######  Theorem 3.

Let $n$ be an odd number. Then for every positive real number $r>n-1,$ the
function $f$ in equation (11) has at most $n-1$ zeros in $(0,\infty).$

###### Proof.

Consider the function $g$ defined as

$g(x)=f(x)\prod_{j=1}^{n}(x+p_{j}).$ (12)

Expanding the product, we can write

$\displaystyle g(x)$
$\displaystyle=\alpha_{0}+\alpha_{1}x+\cdots+\alpha_{n-1}x^{n-1}$
$\displaystyle\ +\beta_{0}x^{r}+\beta_{1}x^{r+1}+\cdots+\beta_{n-1}x^{r+n-1}.$
(13)

The function $g$ can be written as

$g(x)=x^{r}h_{1}(x)+h_{2}(x),$

where

$h_{1}(x)=\sum_{i=1}^{n}c_{i}\prod_{j\neq i}(x+p_{j})\textrm{ and
}h_{2}(x)=\sum_{i=1}^{n}c_{i}p_{i}^{r}\prod_{j\neq i}(x+p_{j}).$

Since both $h_{1}$ and $h_{2}$ are Lagrange interpolation polynomials of
degree at most $n-1$ and not all $c_{i}$ are zero, neither of the polynomials
$h_{1}$ and $h_{2}$ is identically zero. Hence $g$ is not identically zero.
Now, consider the function $g_{0}$ defined as

$g_{0}(x)=\sum_{i=1}^{n}c_{i}\frac{x^{r}-p_{i}^{r}}{x-p_{i}}\prod_{j=1}^{n}(x-p_{j}).$

Then a calculation shows that

$\displaystyle g_{0}(x)$
$\displaystyle=-\alpha_{0}+\alpha_{1}x+\cdots+\alpha_{n-2}x^{n-1}-\alpha_{n-1}x^{n-1}$
$\displaystyle\ +\beta_{0}x^{r}-\beta_{1}x^{r+1}+\cdots+\beta_{n-1}x^{r+n-1}.$
(14)

By the Descartes rule of signs ([12], p.46), the number of positive zeros of
$g$ is no more than the number of sign changes in the sequence of coefficients

$\left(\alpha_{0},\alpha_{1},\ldots,\alpha_{n-1},\beta_{0},\beta_{1},\ldots,\beta_{n-1}\right).$

Let this number of sign changes be $s,$ and let $s_{0}$be the number of sign
changes in the coefficients in (14). Since $n$ is odd, we have $s+s_{0}\leq
2n-1.$ We know that $g_{0}$ has at least $n$ positive zeros
$p_{1},\ldots,p_{n}.$ So, $s_{0}\geq n,$ and hence $s\leq n-1.$ Hence $g$ has
at most $n-1$ positive zeros, and therefore so does $f.$

We can deduce the following.

######  Corollary 4.

Let $n$ be an odd number, and let $p_{1},\ldots,p_{n}$ and
$q_{1},\ldots,q_{n}$ be two $n$-tuples of distinct positive numbers. Then for
every $r>n-1,$ the $n\times n$ matrix

$\begin{bmatrix}\frac{p_{i}^{r}+q_{j}^{r}}{p_{i}+q_{j}}\end{bmatrix}$ (15)

is nonsingular. So, in particular if $n$ is odd, then for every $r>n-1,$ the
matrix $K_{r}$ is nonsingular.

###### Proof.

If the matrix (15) is singular, then there exists a nonzero tuple
$(c_{1},\ldots,c_{n})$ such that

$f(x)=\sum_{j=1}^{n}c_{j}\frac{x^{r}+q_{j}^{r}}{x+q_{j}}$

has at least $n$ zeros $p_{1},\ldots,p_{n}.$ But this is not possible by
Theorem 3. So, the matrix (15) and hence, the matrix $K_{r}$ is nonsingular
for all odd $n$ and $r>n-1.$

We complete the proof of Theorem 1 using a ”snaking” process$:$ the validity
of the theorem is extended by alternatively increasing $n$ and $r.$ For any
positive numbers $p$ and $q$ and any real $r,$ we have

$\frac{p^{r}+q^{r}}{p+q}=p^{r-1}-p\frac{p^{r-2}+q^{r-2}}{p+q}q+q^{r-1}.$

This gives us the identity

$K_{r}=D^{r-1}E-DK_{r-2}D+ED^{r-1},$ (16)

where $D$ is the diagonal matrix $\text{\rm diag}(p_{1},\ldots,p_{n})$ and $E$
the matrix all whose entries are one. For $1\leq j\leq n,$ let
$\mathcal{H}_{j}$ be the subspace of $\mathbb{C}^{n}$ defined as

$\displaystyle\mathcal{H}_{j}$ $\displaystyle=\left\\{x:\sum x_{i}=0,\sum
p_{i}x_{i}=0,\ldots,\sum p_{i}^{j-1}x_{i}=0\right\\}$
$\displaystyle=\left\\{x:Ex=0,EDx=0,\ldots,ED^{j-1}x=0\right\\}.$

Evidently, $\textrm{dim}\,\mathcal{H}_{j}=n-j$ and
$\mathcal{H}_{j+1}\subset\mathcal{H}_{j}.$ It will be convenient to use the
notation$K_{r}^{(n)}$ to indicate an $n\times n$ matrix of the type $K_{r}.$
When the superscript $n$ is not used, it will be understood that a statement
about $K_{r}$ is true for all $n.$ Recall that $K_{r}$ is known to be positive
definite for $0<r<1.$ The relation (6) then shows that it has the same
property for $-1<r<0.$ $K_{0}$ is the Cauchy matrix
$\begin{bmatrix}\frac{2}{p_{i}+p_{j}}\end{bmatrix}$ and is positive definite.
Thus $K_{r}$ is positive definite for $-1<r<1.$ Now let $1<r<3.$ Then
$-1<r-2<1.$ Using the identity (16) we see that if $x$ is a nonzero vector in
$\mathcal{H}_{1},$ then

$\langle x,K_{r}x\rangle=-\langle Dx,K_{r-2}Dx\rangle<0.$

So, the matrix $K_{r}$ is conditionally negative definite and has at least
$n-1$ negative eigenvalues. Since all entries of $K_{r}$ are positive, it has
at least one positive eigenvalue. Thus

$\text{\rm In}\,K_{r}=(1,0,n-1),$ (17)

for $1<r<3.$ By Corollary 4, $K_{r}^{(3)}$ is nonsingular for $r>2.$ So,
$\text{\rm In}\,K_{r}^{(3)}$ does not change for $r>2.$ This shows that
$\text{\rm In}\,K_{r}^{(3)}=(1,0,2)$ for all $r>1.$ So, the theorem is
established when $n=3.$ Now let $n>3$ and $3<r<5.$ Using the identity (16) and
the case $1<r<3$ of the theorem that has been established we see that if $x$
is a nonzero vector in $\mathcal{H}_{2},$ then $\langle x,K_{r}x\rangle>0.$
So, the matrix $K_{r}$ has at least $n-2$ positive eigenvalues. By the $n=3$
case already proved, we know that $K_{r}$ has a $3\times 3$ principal
submatrix with two negative eigenvalues. So, by Cauchy’s interlacing
principle, $K_{r}$ must have at least two negative eigenvalues. We conclude
that

$\text{\rm In}\,K_{r}=(n-2,0,2),\textrm{ if }3<r<5.$ (18)

In particular, this shows that $\text{\rm In}\,K_{r}^{(5)}=(3,0,2)$ if
$3<r<5,$ and since $K_{r}^{(5)}$ is nonsingular for $r>4,$ it has the same
inertia for all $r>3.$ So, Theorem 1 is established when $n=5.$ Next consider
the case $n=4.$ Let $r>3.$ The matrix $K_{r}^{(4)}$ has a principal submatrix
$K_{r}^{(3)}$ whose inertia is $(1,0,2).$ So, by the interlacing principle
$K_{r}^{(4)}$ has at least two negative eigenvalues. On the other hand
$K_{r}^{(4)}$ is a principal submatrix of $K_{r}^{(5)}$ whose inertia is
$(3,0,2).$ So, again by the interlacing principle $K_{r}^{(4)}$ has at least
two positive eigenvalues. Thus $\text{\rm In}\,K_{r}^{(4)}=(2,0,2)$ for all
$r>3,$ and Theorem 1 is established for $n=4.$ This line of reasoning can be
continued. Use the space $\mathcal{H}_{3}$ at the next stage to go to the
interval $5<r<7.$ Then use the established case $n\leq 5$ to extend the
validity of the theorem to first the case $n=7,$ and then $n=6.$

##  3\. Remarks

  1. 1.

In Theorem 1.1(v) of [4] it was shown that all nonzero eigenvalues of $L_{r}$
are simple. We have not been able to prove a corresponding statement for
$K_{r}.$

  2. 2.

An $n\times n$ real matrix $A$ is said to be strictly sign-regular (SSR) if
for every $1\leq k\leq n,$ all $k\times k$ sub-determinants of $A$ are nonzero
and have the same sign. If this is true for all $1\leq k\leq m$ for some
$m<n,$ then we say $A$ is in the class $\textrm{SSR}_{\mbox{m}}.$See [9] for a
detailed study of such matrices. In [4] it was shown that the matrix $L_{r}$
is in the class $\textrm{SSR}_{\mbox{r}}$ if $r=1,2,\ldots,n-1,$ and in the
class SSR for all other $r>0.$ This fact was then used to prove the simplicity
of nonzero eigenvalues of $L_{r}.$

Let $n=4$ and consider the matrix $K_{3}(1,2,5,10).$ It can be seen that the
leading $2\times 2$ principal subdeterminant of this matrix is $-5,$ while the
determinant of the top right $2\times 2$ submatrix is $35.$ So this matrix is
not in the class $\textrm{SSR}_{\mbox{2}}.$

  3. 3.

Let $p_{1}<p_{2}$ and $q_{1}<q_{2}$ be two ordered pairs of distinct positive
numbers such that $\\{p_{1},p_{2}\\}\cap\\{q_{1},q_{2}\\}$ is nonempty. With a
little work it can be shown that the determinant of the $2\times 2$ matrix
$\begin{bmatrix}\frac{p_{i}^{r}+q_{j}^{r}}{p_{i}+q_{j}}\end{bmatrix}$ is
positive if $0<r<1$ and negative if $r>1.$ Using this one sees that for $n=3$
and $r\neq 1,$ the matrix $K_{r}$ is SSR.

  4. 4.

There is a curious and intriguing connection between the inertia of $K_{r}$
and that of another family. For $r\geq 0,$ let $B_{r}$ be the $n\times n$
matrix

$B_{r}=\begin{bmatrix}|p_{i}-p_{j}|^{r}\end{bmatrix}.$ (19)

This family has been studied widely in connection with interpolation of
scattered data and splines. The inertias of these matrices were studied by
Dyn, Goodman and Micchelli in [8]. In view of our theorem their result can be
stated as

$\text{\rm In}\,B_{r}=\text{\rm In}\,K_{r+1}$ (20)

for all $r\geq 0.$ It will be good to have an understanding of what leads to
this remarkable equality. By Sylvester’s law (20) is equivalent to saying that
$B_{r}$ and $K_{r+1}$ are congruent. In a recent work [11] the authors
construct an explicit congruence between these two matrices, thus providing an
alternative proof of (20).

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