gauge theory, Scalar $\equiv$ massive spin-0 running in the loop, Vector
$\equiv$ massive spin-1 running in the loop, $\text{m}\mathcal{N}{=}8\equiv$
massive $\mathcal{N}=8$ supermultiplet in the loop. The new data points
further emphasize the main observation of Ref. Bern:2021ppb , that explicit
data lies on small ‘theory islands’ in the space allowed by unitarity,
causality, and crossing constraints (this larger space is not indicated in
Fig. 1 and is the red-shaded region in Fig. 9 of Ref. Bern:2021ppb ).

###  6.2 Sample data in $D=6$

To showcase some features of our amplitude data, we generate similar data
plots for graviton polarizations that are outside the four-dimensional
helicity setup. To this end, we first consider scattering of gravitons in
$D=6$ with the polarization choice
$\mathcal{M}(1_{6^{-}}2_{6^{-}}3_{6^{+}}4_{6^{+}})$ where all polarizations
are outside the four-dimensional subspace. (See Eq. (6) for our conventions of
the graviton polarization states.) We consider the coefficients of the
$1/m^{2}$ terms in Fig. 2 which are polynomials in $s,t$ of degree four. The
particular helicity choice renders the amplitude $t\leftrightarrow u$
symmetric and we denote the amplitude coefficients by their corresponding
monomial in the Mandelstams. Unlike for the $D=4$ examples considered
previously, it is not always possible to factor out some overall powers of
Mandelstams.

Figure 2:  The $D=6$ EFT data for various models for $a[s^{3}t]/a[s^{4}]$ and
$a[s^{2}t^{2}]/a[s^{4}]$. A line with slope $3/2$ is added to guide the eye.
Here $a[x]$ stands for the coefficient of the monomial $x$ in the Taylor
expansion of the amplitude.

It is fascinating to observe that, similarly to four-dimensional theories, the
data points of the various models lie on an almost straight line and their
spread from the line is much smaller than the extend of the line itself,
giving us a concrete first example of small theory islands beyond $D=4$.

We should, however, stress that not all extra-dimensional data falls on such
perfect lines. To see this, let us investigate a three-dimensional section of
the coefficient space at mass-level $1/m^{6}$ where we have degree-6
polynomials in $s$ and $t$. We summarize our results in Fig. 3. As seen in a
rotated viewpoint shown in Fig. 4, this data essentially lies in a plane.
While it appears that the virtual scalar EFT lies somewhat off a line formed
by the other models, it is difficult to assess the broader significance of
this departure vis-à-vis the parameter space allowed by causality, unitarity
and crossing constraints, which is currently not known beyond $D=4$.

Figure 3:  The $D=6$ EFT data for various models for the ratios of low-energy
amplitude coefficients $a[s^{5}t]/a[s^{6}]$, $a[s^{4}t^{2}]/a[s^{6}]$, and
$a[s^{3}t^{3}]/a[s^{6}]$. We add a straight line to guide the eye. As
illustrated in Fig. 4 from a different viewpoint the data essentially lie in a
plane.  Figure 4:  The same data points and line of Fig. 3 from a different
viewpoint that demonstrates that the data essentially lie in a plane.

###  6.3 Sample data in $D=10$

We proceed to analyze the data provided by perturbative calculations in
$D=10$, which is an interesting dimension from a superstring perspective. We
observe that, as in lower dimensions, the ratios of four-graviton amplitude
coefficients again lie on a remarkably thin island. As mentioned in the
introduction, this theoretical data should provide crucial guidance for future
dispersive analyses analogous to those carried out in $D=4$. Here, we only
plot one particular section through our data in $D=10$ which could eventually
interplay with the search for string theory via the analysis of graviton
scattering. For concreteness, we consider the amplitude
$\mathcal{M}(1_{6^{-}}2_{10^{-}}3_{10^{+}}4_{6^{+}})$.

Taking the large-mass expansion and evaluating all tensor structures for the
specified graviton-polarization choice we collect, for example, the expansion
coefficients similar to the $k=4$ coefficients in $D=4$. Notably, in $D=10$ it
no longer holds that we can factor out an overall helicity-dependent
polynomial of the Mandelstam invariants. Therefore, the formerly $k=4$
amplitude coefficients are associated to honest degree-8 polynomials in $s,t$.
In the particular example we discuss, even though we do not naively have
crossing symmetry due to the polarization choice, the only nontrivial
contractions of polarization vectors that survive are
$\varepsilon_{2}\cdot\varepsilon_{3}$ and
$\varepsilon_{1}\cdot\varepsilon_{4}$ which is left invariant under
$2\leftrightarrow 3$ or $1\leftrightarrow 4$. At the level of the polynomials
in Mandelstam invariants, this leaves three independent degrees of freedom
that we can in general bound from unitarity, causality, crossing and Regge-
behavior considerations. The three independent coefficients are associated to
the $s^{8}$, $s^{6}t^{2}$ and $s^{4}t^{4}$ terms respectively and their ratios
are depicted in Fig. 5.

Figure 5: The $D=10$ EFT data for various models for the ratios of low-energy
amplitude coefficients $a[s^{6}t^{2}]/a[s^{8}]$ and $a[s^{4}t^{4}]/a[s^{8}]$.
A line with slope $9/2$ is added to guide the eye.

The data lie on an almost straight line of approximate slope $9/2$. We leave
to future work to establish the appropriate bounded regions of allowed
parameter space from unitarity and causality constraints. However, the
similarity between Figs. 1 and 5 suggests that many of the interesting
features of the four-dimensional theory islands survive in higher dimensions
as well.

Figure 6: The $D=10$ EFT data with four-dimensional external polarizations
$4_{\pm}$. Here we follow the conventions of Eq. (28). A line with slope 3/2
is added to guide the eye.

Recently, the projective bounds between independent Wilson coefficients of the
same mass-level $k$ have been generalized to extremely interesting bounds of
Wilson coefficients against e.g. Newton’s constant Caron-Huot:2022ugt ;
Chiang:2022jep . Here, we chose to plot projective data points due in part to
the surprising observation that Fig. 1 in $D=4$ does not change significantly
compared to Fig. 6 in $D=10$, where in both cases we evaluate the amplitudes
for polarizations corresponding to four-dimensional helicity states.
Specifically, only the “Vector” data point is $D$-dependent, and this
dependence is solely due to the $D_{s}$ appearing in Eq. (24). The net effect
of this $D$-dependence is that the “Vector” data point is closer to the
“Scalar” in Fig. 6 compared to Fig. 1. While we do not spell out the details,
this relative uniformity of the data across dimensions can be understood from
the analysis of dimension-shifted scalar integrals. This is consistent with
the intuition that extra-dimensional momenta in the loop can be thought of as
a Kaluza-Klein mass which we effectively integrate over. This integral over
the mass then drops out from the projective data.

##  7 Conclusions and Outlook

In this work we studied explicit perturbative and nonperturbative models for
UV sensible four-graviton scattering amplitudes and their respective low-
energy expansions. In particular, as a first example, we studied a
nonperturbative $\mathcal{N}=1$ supersymmetric gauge theory also coupled to
gravity in four spacetime dimensions. Moreover, in $D$ dimensions we
considered tree-level four-graviton amplitudes in string theory and computed
minimally-coupled one-loop four-graviton amplitudes with massive matter
circulating in the loop.

The crucial output is the low-energy expansion of these amplitudes. As in the
four-dimensional EFTs considered in Ref. Bern:2021ppb , the low-energy
coefficients of our data populate rather small theory islands compared to the
naive expectation that consistent theories should fill out the full space of
allowed low-energy couplings. Unlike the four-dimensional case where this
feature was attributed to low-spin dominance, in general spacetime dimensions
we do not currently have the same level of understanding. Nonetheless, our
explicit analysis suggests that similar mechanisms are at work and it would be
extremely interesting to further explore it by studying the partial-wave
decomposition of our higher-dimensional amplitudes. We also anticipate that
our explicit data will serve as a useful guide for any upcoming analysis of
dispersive bounds on the low-energy Wilson coefficients. Further data can be
found by exploiting the results of Ref. Edison:2021ebi , in which one-loop
amplitudes in theories deformed by operators induced by integrating out
massive string states are computed. Similarly to the discussion in section
5.1, the states circulating in the loop can be rendered massive through
Kaluza-Klein reduction; then, the resulting amplitudes are interpreted as
those of an EFT of compactified string theory valid at scales $p^{2}\sim
m_{\text{KK}}^{2}\ll(\alpha^{\prime})^{-2}$, in the same spirit as the
discussion in section 3.

There are several further interesting directions to pursue. For one, beyond
$D=4$, gravitational scattering amplitudes are no longer plagued by infrared
singularities that hamper obtaining a number of four-dimensional bounds from
various approaches and can show up in the form of IR-logarithms even beyond
the forward limit bounds Caron-Huot:2022ugt . Gravitational scattering in
higher dimensions is also interesting in light of the recent attempts to find
string theory Guerrieri:2021ivu  from an S-matrix bootstrap point of view. It
would be very interesting to narrow down the range of possible extensions of
gravitational UV completions in 10 spacetime dimensions beyond string theory.
One could also combine the scattering matrix for different graviton states in
higher dimensions into a matrix, on which nontrivial bounds can be determined,
analogous to the ones found for different helicity configurations in four
dimensions Bern:2021ppb .

Additionally, the data presented here is primarily deduced from conventional
gravitational scattering amplitudes, although we also presented a first
example of data deduced from nonperturbative amplitudes. Recently, certain
more exotic amplitudes with “accumulation point spectra” have gained some
attention in e.g. Refs. Figueroa:2022onw ; Huang:2022mdb  and likewise in the
model of Appendix D of Ref. Bern:2021ppb  that in turn was inspired by Ref.
Caron-Huot:2020cmc . It is interesting to note that while the accumulation-
point model of Ref. Bern:2021ppb  was designed to violate low-spin dominance,
the associated low-energy EFT still belongs to the same theory island as the
more conventional string- and field-theory data. It would be interesting to
study the ultimate fate of accumulation-point amplitudes and determine whether
or not they should be thought of as physical UV completions.

Along similar lines, it would be interesting to think about additional models
that UV-complete gravitational scattering. Perhaps the most interesting new
classes of theories involves nonperturbative physics. Here, we studied only a
first example. It is possible to carry out a controlled non-perturbative
analysis in the context of the AdS/CFT correspondence Maldacena:1997re ;
Witten:1998qj ; Gubser:1998bc , generalized as in e.g. Ref. Polchinski:2000uf
or with a simple cutoff in the transverse direction Polchinski:2001tt , to
describe a confining boundary theory. Indeed, bulk AdS5 supergravity can be
used to find the boundary four-dimensional correlation functions of e.g. four
stress tensors, which in turn can be interpreted as the four-graviton off-
shell Green’s functions. Fourier-transforming to boundary momentum space
allows on-shell conditions to be imposed and leads to the boundary four-
graviton amplitude due to virtual nonperturbative matter. Naively, the
contribution of one bulk-exchange diagram to the amplitude is, up to a
polynomial helicity-dependent factor, a function of the corresponding
Mandelstam invariant. It would be very interesting to explore the consequences
of these polynomial factors on the Taylor coefficients of the expansion of the
amplitude at large confinement scale.666At high energies the scattering
process is localized in AdS space and given, up to external-state factors, by
tree-level string theory amplitudes Polchinski:2001tt . This suggests—but does
not prove—that we may expect certain similarities between the properties of
flat-space string-theory Taylor coefficients and those of the boundary
graviton amplitudes.

In conclusion, placing bounds on gravitational scattering and analyzing
explicit data in four spacetime dimensions and beyond should give us a
fruitful probe of possible extensions of Einstein gravity. A key question is
whether we can constrain sensible theories to live on islands as small as
those suggested by the explicit theoretical data.

### Note added:

While completing this manuscript, we were informed by Simon Caron-Huot, David
Simmons-Duffin, Julio Parra-Martinez, and Yue Zhou about their study of
“Graviton partial waves and causality in higher dimensions” Caron-
Huot:2022toAppear . It would be interesting to compare the explicit data
presented here to the bounded regions derived from the dispersive arguments
and study possible low-spin dominance explanations of the small theory
islands.

## Acknowledgments:

We are grateful to Simon Caron-Huot, David Simmons-Duffin, Julio Parra-
Martinez, and Yue Zhou for pointing out their upcoming work on higher-
dimensional spinning partial waves to us prior to publication Caron-
Huot:2022toAppear . We thank Petr Kravchuk and Alexander Zhiboedov for
enlightening discussions and collaboration on related topics. We further wish
to thank Callum Jones, Mikhail Solon and Fei Teng for fruitful conversations.
Z.B, E.H. and D.K. are supported by the U.S. Department of Energy (DOE) under
Award Number DE-SC0009937. R.R. is supported by the U.S. Department of Energy
(DOE) under award number DE-SC00019066. We also are grateful to the Mani L.
Bhaumik Institute for Theoretical Physics for support.

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Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations.

§ INTRODUCTION

Consider the vector space of $n\times n$ symmetric matrices, $\R^{n(n+1)/2}$. In this vector space, a semidefinite programming (SDP) problem involves minimizing a linear function over the intersection of the convex cone of positive semidefinite matrices with an affine subspace. In particular,

minimize     CX_F

(SDP)     subject to     A_iX_F = b_i       i = 1,2,…n

X ∈n  .

where $\S{n} \define \set{X \in \R^{n\times n}}{X = X^\top,\, \y^\top X\y \geq 0~~\forall~~\y \in \R}$$\S{n} := \set{X \in \Rsym{n}}{X \succeq 0}$ denotes the cone of all $n\times n$ real symmetric positive semidefinite matrices. SDPs have been studied extensively in optimization literature, and have been used as convex relaxations of several non-convex optimization problems [Goemans and Williamson, 1994, Oustry et al., 2022, Sojoudi and Lavaei, 2014, Dey et al., 2022]. As large-scale SDPs are difficult to solve computationally, a general approach is to consider relaxations of the problem where instead of the variable $X$, only $k \times k$ principal submatrices of $X$, for $2 \leq k < n$ are constrained to be positive semidefinite [Kim and Kojima, 2001, Kocuk et al., 2016].
($k$-PSD closure [Blekherman et al., 2022])
Given positive integers $n$ and $k$ where $2 \leq k \leq n$, the $k$-PSD closure $\Sminors{n}{k}$ is the set of all $n \times n$ symmetric real matrices where all $k \times k$ principal submatrices are PSD (positive semidefinite).

In certain cases, these relaxations yield near accurate solutions to the (SDP) problem [Kocuk et al., 2016, Sojoudi and Lavaei, 2014, Dey et al., 2022]. The question pertaining to quantifying the closeness of approximation of $\calS^{n,k}_+$ and $\calS^{n}_+$ has been explored recently [Blekherman et al., 2022, Blekherman et al., 2022]. These works primarily utilize a projection metric to quantify the closeness of approximation. To further improve the projection metric bounds of Blekherman et al., 2022, in a subsequent work Blekherman et al. [Blekherman et al., 2022] spectral analysis of $\calS^{n,k}_+$.  The convex cone of eigenvalues of matrices in $\calS^{n,k}_+$ is called the hyperbolicity cone and its properties are studied in [Blekherman et al., 2022, Kozhasov, 2023]. Using certain properties of the hyperbolicity cone, Blekherman et al., 2022 improve upon the Frobenius distance bounds of an earlier work [Blekherman et al., 2022].

A different direction of research studies the number of $k\times k$ psd cones needed to approximate the $\calS^n_+$ positive semidefinite cone [Fawzi, 2019, Fawzi, 2021, Song and Parrilo, 2023]. It has been established that the number of cones needed to approximate the $\calS^n_+$ cone is exponential if $k$ is small compared to $n$. Consequently, these works conclude that the $k$-PSD closure relaxations can be arbitrarily bad. There have been some explorations in the literature relating to the exactness of these relaxations under sparse affine constraints [Grone et al., 1984, Fukuda et al., 2001, Sojoudi and Lavaei, 2014, Vandenberghe et al., 2015].

Our motivation to study these relaxations of the positive semidefinite cone primarily stems from the following: Even though, in the worst case scenario, a $k-$ PSD closure $(k > 1)$ yields an arbitrarily bad relaxation  of the PSD cone, the two cones do share some common faces/generators. In that case, given a linear objective function, understanding the geometry of these relaxations might provide some insight into the quantifying the quality of the relaxation $\Sminors{n}{k}$ for $\S{n}$, which may not be arbitrarily bad. In fact, it may give a geometric insight into why some relaxations are exact.Our motivation stems from the fact that understanding the geometry of these relaxations might provide some insight into the quantifying the closeness of the relaxation $\calS^{n,k}_+$ to $\calS^n_+$ with respect to some lineart objective functionproblem of why these relaxations i.e, $\calS^{n,k}_+$ give near optimal solutions. Specifically, we study the extreme rays of the relaxed convex cones $\calS_+^{n,k}$ and try to understand the number of new generators that are added to the generators of the PSD cone, as $k$ varies. Gouveia et al. [Gouveia et al., 2022] characterize the extreme rays of $\calS_+^{n,n-1}$ and provide characterization of the dual cone of $\calS_+^{n,k}$ called the factor-width cone. Additionally, they completely characterize the extreme rays of $\calS^{4,3}_+$. The dual cone of $\calS^{n,k}_+$ is called the factor-width $FW^n_k$ cone of factor-width $k$. A structural result about NLS, non-singular locally singular, matrices of $\calS^{n,k}_+$ is proven in [Blekherman et al., 2022]. In the following, we characterize the extreme rays of $\mathcal{S}_+^{n,2}$ and provide a different proof for characterization of the extreme rays of $\mathcal{S}_+^{n,n-1}$ to the one already given in [Gouveia et al., 2022].

The manuscript is arranged as follows: Section <ref> we state our  results. In Section <ref> we define the notation, definitions and some preliminary lemmas. Finally in Section <ref> we provide proofs of stated results

§ MAIN RESULTS

[prop:extrays] For $\mathbf{X} \in \Rsym{n}$, the following hold:

* [prop:extrays1] $\X$ spans an extreme ray of $\Snk{n}{k}$ if all $k \times k$ principal submatrices of $\X$ are rank-1 and positive semidefinite.

* [prop:extrays2] $\X$ spans an extreme ray of $\Snk{n}{2}$ if and only if all $2 \times 2$ principal submatrices of $\X$ are rank-1 and positive semidefinite.

* [prop:extrays3] $\X$ spans an extreme ray of $\Snk{n}{n-1}$ if and only if all $(n-1) \times (n-1)$ principal submatrices of $\X$ are positive semidefinite and

* either rank-1,

* or rank-$(n-2)$ such that $\det(\X) < 0$.

The corollary follows from Theorem <ref>.

For $n\geq 3$ and $2\leq k\leq n-1$, \(G(n,k) = \dfrac{k}{k-1}\mathbb{I} - \dfrac{1}{k-1}\mathbf{1}\mathbf{1}^{\top}\) spans an extreme ray of \(\Sminors{n}{k}\).

A related structure termed as non-singular locally singular ($\emph{NLS}$) matrices was introduced by Blekherman et al., 2022 as following:
[NLS Matrix [Blekherman et al., 2022]]
A matrix $\M\in \calS^{n,k}_+$ is (NLS) for $n\geq 5$ and $3 \leq k\leq n-2$ if all the $k\times k$ principal minors are zero and $\det(\M)\neq 0$.

NLS matrices are diagonally congruent to $G(n,k)$ matrices which are used derive distance bounds between $\calS^{n,k}_+$ and $\calS^n_+$ [Blekherman et al., 2022]. The following result with respect to NLS matrices follows from Corollary <ref>.

An NLS matrix $\M \in \calS^{n,k}_+$, for $n\geq 5$ and $3\leq k\leq n-2$, spans an extreme ray of $\calS^{n,k}_+$.

§ PRELIMINARIES

§.§ Notation

$\R^n$ denotes the $n-$dimensional Euclidean space and the set $\Rsym{n} \define \set{X \in \R^{n\times n}}{X = X^\top}$ defines the set of $n\times n$ symmetric real matrices. By $[n]$, we denote the index set $\{1,2,\ldots,n\}$. Additionally, $\Q \in \Rsym{n}$ is a positive semidefinite matrix, denoted by $\Q \succeq 0$, if and only if $\Q \in \S{n}$. For $\M \in \Rsym{n}$ we denote by $\M_I$, the $k\times k$ principal sub-matrix of $\M$ indexed by columns and rows $I \subseteq [n]$, $|I| = k$. Analogously, for $\ve \in \R^n$, $\ve_J$ denotes the subvector of $\ve$ corresponding to the index set $J\subseteq [n]$. Additionally, for $i, j \in [n]$, $\M_{ij}$ represents the submatrix of $\M$ indexed by rows $[n]\sm\{i\}$ and columns $[n]\sm\{j\}$. It follows that, for some $i\in [n]$, $\M_{[n]\sm\{i\}}$, the $(n-1)\times (n-1)$ principal sub-matrix of $\M$ can also be represented as $\M_{ii}$. The element of a matrix $\M$ indexed by $i^{th}$ row and $j^{th}$ column is represented as $m_{ij}$. Similarly, $v_j$ is the $j^{th}$ element of the vector $\ve_j$. We denote by $\tr(\M)$ and $\adj(\M)$ the trace and adjugate of the matrix $\M$, respectively. $\mathbb{I}_{n}$ and $\mathbf{1}_n$ represent the $n \times n$ identity matrix and the $n$ dimensional vector of all ones, respectively. $\sign$ represents signum function. $\langle\,. \,,\,.\rangle_F$ denotes Frobenius inner product on the set of symmetric matrices.

§.§.§ Preliminary Lemmas

[Matrix Determinant Lemma, [Harville, 1998]]
Given a matrix $\M \in \Rsym{n}$ and $\u,\ve \in \R^n$,
$$\det(\M + \u{\ve}^\top) = \det(\M) + \tr(\adj(\M)~\u{\ve}^\top)$$

Consider $\M\in \R^{n\times n}$. If $\exists ~i \in [n]$ such that $\rk(\M_{ii}) = r$. Then $\rk(\M) \geq r$.

Consider $\M\in \mathcal{S}^n$. Let \(J = [n]\sm\{i\}\) and $\m^{(i)}$ be the $i^{th}$ row of $\M$. Then
$$\det(\M) = m_{ii}\det(\M_{J}) - \m^{(i)\top}_{J} \adj(\M_{J})~\m_{J}^{(i)}.$$

Without loss of generality, let $i = 1$. From Laplace's equation, by expanding along the first row we have,

\begin{align}\label{prop_eq:6_1}
\det(\M) &= \sum_{j=1}^n (-1)^{(1+j)} m_{1j}\det(\M_{1j}) = m_{11}\det(\M_{11})+\sum_{j=2}^n (-1)^{(1+j)} m_{1j}\det(\M_{1j}).
\end{align}

Let $\m^1$ be the first column of the matrix $ \M $ and $ J \define [n]\sm\{1\} $.
One can obtain the sub-matrix $\M_{1j}$ from $\M_{11}$ by performing two operations on $\M_{11}$.

* The first operation is to replace $(j-1)^{th}$ column of $ \M_{11}$ with $\m_{J_1}^1$. Let the matrix obtained from the first operation be $\M'_{1j}$.

Then $\M'_{1j}$ is same as $\M_{11}$ except the $(j-1)^{th}$ column. Therefore $(j-1)^{th}$ row of $\adj(\M_{11})$ and $\adj(\M'_{1j})$ are the same.

\begin{align*}
\adj(\M'_{1j})\M'_{1j} = \det(\M'_{1j}) \mathbb{I}_{n-1}.
\end{align*}

Here $ \mathbb{I}_{n-1} $ is the $ (n-1)\times (n-1) $ identity matrix.
Let the $(j-1)^{th}$ row of $\adj(\M_{11})$ be $(\mathbf{a}^{j-1})^{\top}$. We have,

\begin{equation}\label{prop_eq:6_2}
\det(\M'_{1j}) = (\mathbf{a}^{j-1})^{\top}\m_{J_1}^{1}.
\end{equation}

* The second operation is to sequentially interchange columns of the matrix $\M'_{ij}$ i.e, interchange the $(j-1)^{th}$ column with the $ (j-2)^{th} $, then the $ (j-2)^{th} $ with the $ (j-3)^{th} $, and so on till $ \m_{J_1}^1 $ is the first column. Each column interchange changes the sign of the determinant by $-1$. As there are $ j-2 $ column interchanges we have,

\begin{equation}\label{prop_eq:6_3}
\det(\M_{1j}) = (-1)^{j} \det(\M'_{1j}).
\end{equation}

From (<ref>) and (<ref>), we obtain

\begin{equation}\label{prop_eq:6_4}
\det(\M_{1j}) = (-1)^j \mathbf{a}_{j-1} \m_{J_1}^1.
\end{equation}

Combining (<ref>) and (<ref>),

\begin{align*}
\det(\M) &= m_{11}\det(\M_{J}) +\sum_{j=2}^n (-1)^{1+j}\m_{1j} \det(\M_{1j})\\
&= m_{11}\det(\M_{J}) + \sum_{j=2}^n (-1)^{1+j}\m_{1j} (-1)^j (\mathbf{a}^{j-1})^{\top}\m^{1}_{J}\\
&= m_{11}\det(\M_{J})- (\m_{J}^{1})^{\top} \adj(\M_{11})\m^{1}_{J}.
\end{align*}

Consider $\M\in\mathcal{S}^n$. If $\mathsf{rank}(\M) = n-1$ then $\mathsf{rank}(\adj(M)) = 1$. Furthermore, if $\M\in \mathcal{S}_+^n$ then $\adj(\M) \succeq \bz$ i.e, $\adj(\M) = \u\u^{\top}$ for some $\u\in\R^n$.

Let $J_{1},J_{2},\ldots,J_{k}\subset[n]$$J_{1},J_{2},\ldots,J_{k}\subset[n]$ where $|J_i|=n-1~\forall i \in [k]$. If $k\leq n-1$ and $J_i\neq J_j\,\, \forall i\neq j$,
be pairwise distinct subsets of $[n]$ with $|J_i| = n-1$.
Then $\displaystyle \bigcap_{i=1}^k J_{i} \neq \emptyset$.

As $|J_i| = n-1$ it can be written as we can write $J_i = [n]\backslash\{j_i\}$ for some $j_i \in [n]$. Let $J = [n]\backslash\{j_1,j_2,\ldots,j_k\}$. Observe that $J\subset J_{i}\,\,\forall i\in [k]$. As $|J|\geq 1$, we have that $J\subseteq \displaystyle \bigcap_{j=1}^k J_{i} \neq \emptyset$.

§ MAIN RESULTS

[prop:extrays] For $\mathbf{X} \in \Rsym{n}$, the following hold:

* [prop:extrays1] $\X$ spans an extreme ray of $\Snk{n}{k}$ if all $k \times k$ principal submatrices of $\X$ are rank-1 and positive semidefinite.

* [prop:extrays2] $\X$ spans an extreme ray of $\Snk{n}{2}$ if and only if all $2 \times 2$ principal submatrices of $\X$ are rank-1 and positive semidefinite.

* [prop:extrays3] $\X$ spans an extreme ray of $\Snk{n}{n-1}$ if and only if all $(n-1) \times (n-1)$ principal submatrices of $\X$ are

* either rank-1 and positive semidefinite

* or rank-$(n-2)$ such that $\det(\X) < 0$.

The corollary follows from Theorem <ref>.

\(G(n,k) = \dfrac{k}{k-1}\mathbb{I} - \dfrac{1}{k-1}\mathbf{1}\mathbf{1}^{\top}\) spans an extreme ray of \(\Sminors{n}{k}\).

Blekherman et al. [Blekherman et al., 2022] introduced non-singular locally singular ($\emph{NLS}$) matrices as following:
[NLS Matrix [Blekherman et al., 2022]]
A matrix $\M\in \calS^{n,k}_+$ is (NLS) for $n\geq 5$ and $3 \leq k\leq n-2$ if all the $k\times k$ principal minors are zero and $\det(\M)\neq 0$.

NLS matrices are diagonally congruent to $G(n,k)$ matrices which are used derive distance bounds between $\calS^{n,k}_+$ and $\calS^n_+$ [Blekherman et al., 2022].

An NLS matrix $\M \in \calS^{n,k}_+$, for $n\geq 5$ and $3\leq k\leq n-2$, spans an extreme ray of $\calS^{n,k}_+$.

§ PROOFS

§.§ Proof Strategy

The idea for characterizing the extreme rays of $\calS^{n,2}_+$ is to construct a perturbation of the matrix $\M \in \calS^{n,2}_+$ if any of the $2\times 2$ principal sub-matrix has rank $2$. And if a $\M\in \calS_+^{n,2}$ such that all its $2\times 2$ principal minors are rank 1 then the result follows.

The idea for characterizing the extreme rays of $\mathcal{S}^{n,n-1}_+$ is to characterize the ranks of the $(n-1)\times (n-1)$ principal sub-matrices of a $\M \in \mathcal{S}^{n,n-1}_+$ which span an extreme ray. We first eliminate the cases for which the $\M \in \mathcal{S}_+^{n,n-1}$ under given ranks of $(n-1)\times (n-1)$ principal sub-matrices.
The characterization of the $(n-1)\times (n-1)$ principal sub-matrices is divided into as follows,

* There exists at least one $(n-1) \times (n-1)$ principal sub-matrix whose rank is less than or equal to $n-3$. We prove that these matrices do not span an extreme rays.

* There exists at least one $(n-1) \times (n-1)$ principal sub-matrix whose rank is $n-1$. We prove this by constructing a perturbation of $\X$ for $\M \in \calS_+^{n,n-1}$ which satisfies the aforementioned hypothesis.

The final case is if all the $(n-1)\times (n-1)$ principal sub-matrices have rank $n-2$. If all the $(n-1)\times (n-1)$ sub-matrices have rank $n-2$ and $\mathsf{det}(\M) \geq 0$ then this matrix is positive semidefinite matrix of rank $n-2$, therefore it doesn't span an extreme ray of $\mathcal{S}^{n,n-1}_+$.

Finally, we prove that if a matrix $\M\in \calS^{n,n-1}_+$ and its $(n-1)\times (n-1)$ principal sub-matrices have rank $n-2$ and $\mathsf{det}(\M)< 0$ then it spans an extreme ray of $\mathcal{S}^{n,n-1}_+$.

§.§ Proof of Theorem <ref>

Let $\X \in \Snk{n}{k}$ with all $k \times k$ principal submatrices being rank-1 positive semidefinite, and suppose that $\X = \frac{1}{2}(\Y + \mathbf{Z})$ for some $\Y, \mathbf{Z} \in \Snk{n}{k}$. Additionally, let $\mathcal{I}_k$ denote the set of all index subsets of $\{1,2,\ldots, n\}$ of cardinality $k$ and for $I \in \mathcal{I}_k$, $X_I = [x_{ij}]_{i,j \in I}$. For any $I \in \mathcal{I}_k$, we have

\[{\X_I} =
\frac{1}{2}{\Y_I}
\frac{1}{2}{\mathbf{Z}_I}.
\]

As $\X_I, \Y_I$, and $\mathbf{Z}_I$ all belong to $\S{k}$, and $\X_I$ is of rank 1, we see that $\Y_I$ and $\mathbf{Z}_I$ must be scalar multiples of $\X_I$. By iterating $I$ over all elements of $\mathcal{I}_k$, we see that $\Y$ and $\mathbf{Z}$ must be scalar multiples of $\X$.

§.§ Proof of Theorem <ref>

From definition of $\Snk{n}{2}$, all $2 \times 2$ principal submatrices of $\X$ are positive semidefinite. Now, if a particular minor $\begin{bmatrix}
x_{ii} & x_{ij}\\
x_{ij} & x_{jj}
\end{bmatrix}$
has rank 2, i.e., $x_{ii}x_{jj} - x_{ij}^2 > 0$, then $\X = \frac{1}{2}(\Y + \mathbf{Z})$, where $\Y = \X - \varepsilon(F_{ij} + F_{ji})$, and $\mathbf{Z} = \X + \varepsilon(F_{ij} + F_{ji})$ for some sufficiently small $\varepsilon > 0$. Here, $F_{ij}$ is the matrix with 1 in the $(i,j)$-th entry and 0 elsewhere. Sufficiency follows from Theorem <ref>.

§.§ Proof of Theorem <ref>

Consider $\M\in \Sminors{n}{n-1}$ such that $\rk(\M) > 1$. If $\exists~i \in [n]$ such that $\rk(\M_{ii}) < n-2$ then $\M$ does not span an extreme ray of \(\mathcal{S}_+^{n,n-1}\).If one of the $(n-1)\times (n-1)$ principal minors has rank less than $n-2$ then $\M$ does not span an extreme ray of \(\mathcal{S}_+^{n,n-1}\).

Without loss of generality assume $\mathsf{rank}(\M_{11})<n-2$.
The principal sub-matrix $\M_{11}$ and the sub-matrix $\M_{1j}$ have $n-2$ columns in common for $j=2,3,\ldots,n$.
These $n-2$ columns will have a rank of at most $n-3$ and adding a column to these $n-2$ common columns will make the rank of $\M_{1j}$ at most $n-2$.
Since $\det(\M_{1j}) = 0\, \forall j = 1,\ldots,n$, $\det(\M) = 0$. As all the principal minors are non-negative, this implies that $\M \succeq \bz$.
As $ \M\succeq \bz $ and $\mathsf{rank}(\M) >1 $, we can write $ \M $ as a conic combinations of rank-1 psd matrices. Hence $\M$ does not span an extreme ray.

Let $\M \in S_{+}^{n,n-1}$ and all of its \((n-1)\times (n-1)\) principal sub-matrices have rank at least rank $n-2$. If \(\M\) has at least one full rank \((n-1)\times (n-1)\) principal sub-matrix, then $\M$ does not span an extreme ray of \(\mathcal{S}_+^{n,n-1}\).

Observe that as $\M $ has a full rank $ (n-1)\times (n-1) $ principal sub-matrix whose rank is $ n-1 $ this implies that $\mathsf{rank}(\M) = n-1 $ and hence it follows that $\M$ can have at most one zero column. If $\M$ has a zero column, then $\det(\M) = 0$ and consequently $\M$ is positive semidefinite matrix.
As $\M\succeq \bz$  and $\mathsf{rank}(\M) = n-1 $ hence it can be expressed as a conic combination of rank 1 matrices.

Alternatively, if $ \M $ does not have any zero columns then we prove the existence of a rank 1 matrix, $ \varepsilon\u\u^{\top} $ where
$ \u\in\R^n $ and $ \varepsilon\in \R_{++} $, such that $ \M +\varepsilon\u\u^{\top},\M -\varepsilon\u\u^{\top}\in \mathcal{S}_+^{n,n-1} $.
However, $ \M +\varepsilon \u\u^{\top}\in \mathcal{S}_+^{n,n-1} $ since $ \varepsilon\u\u^{\top}\in \mathcal{S}_+^{n,n-1} $ for all $ \varepsilon > 0 $ and $ \u\in \R^n $.
It suffices to show that, there exists $ \varepsilon > 0$ and $ \u \in \R^n\backslash\{\bz\}$ such that $ \M -\varepsilon\u\u^{\top}\in \mathcal{S}_+^{n,n-1} $. Alternatively, it suffices to show that

\begin{equation}\label{prop_eq:7_1}
\sign(\det(\M_H - \varepsilon\u_H\u_H^{\top}))\geq \sign(\det(\M_H)) \,\, \forall H\subset[n] \text{ and } |H|\leq n-1 .
\end{equation}

Let $\mathcal{H}_{fr}\define \{J\subset[n]: \mathsf{rank}(\M_J) = |J|\text{ and } |J|\leq n-1\}$ and $\mathcal{H}_{nr} \define \{J\in[n]: \mathsf{rank}(\M_J)< |J| \text{ and } |J|\leq n-1\}$. From the matrix determinant lemma and (<ref>), we need to show that

\begin{equation}\label{prop_eq:7_4}
\det(\M_{H} - \varepsilon\u_H\u_H^{\top}) = \det(\M_{H}) - \varepsilon\u^{\top}_H\adj(\M_H)\u_H > 0 \quad\forall H\in \mathcal{H}_{fr}, \text{ and}
\end{equation}

\begin{equation}\label{prop_eq:7_5}
\det(\M_{H} - \varepsilon\u_H\u_H^{\top}) = - \varepsilon\u^{\top}_H\adj(\M_H)\u_H =0 \quad \forall H\in \mathcal{H}_{nr}.
\end{equation}

(<ref>) and (<ref>) hold if and only if (<ref>) and (<ref>). Rewriting (<ref>) and (<ref>) we get,

\begin{align}\label{prop_eq:7_2}
\varepsilon < \min_{H \in \mathcal{H}_{fr}}\frac{\det(\M_H)}{\u^{\top}_H\adj(\M_H)\u_H}
\end{align}

\begin{equation}\label{prop_eq:7_3}
\adj(\M_H)\u_H = \bz   ~\forall H \in \mathcal{H}_{nr}.
\end{equation}

To see this, observe that if $\M$ has non-zero columns, and given that $\M\in \mathcal{S}^{n,n-1}_+$, then we have that the diagonal elements of $\M$ are non-zero. This implies that $\{j\}\in \mathcal{H}_{fr}\,\,\forall j \in[n]$. As $\u\neq\bz$, we have that $\exists i\in[n]$ such that $u_i \neq 0$. This implies that for $H=\{i\}$ we have that $u_i\adj(\M_H)u_i$ is non-zero as $\adj(\M_H)=1$. Therefore, the right-hand side term of the inequality (<ref>) is positive and finite.
Hence $\varepsilon $ has to satisfy (<ref>) and $\u$ has to satisfy \eqref{prop_eq:7_3}.
Note that, if $ℋ_nr$ is empty, then one can find $ε>0$ for any $≠̆$ from \eqref{prop_eq:7_2}.
% And also, $H\in \mathcal{H}_{fr}$ any $\u\neq \bz$ works as along it satisfies \eqref{prop_eq:7_2}.

Let $𝒥 {J∈[n]: |J| = n-1 }$ and $ℐ {I∈[n]: 𝗋𝖺𝗇𝗄(_I) = n-2}$. Observe that $𝗋𝖺𝗇𝗄(_J) = n-1   ∀J∈𝒥\ℐ$. If $ℐ = ∅$ then $ℋ_nr$ is empty. To see this, observe that $_J$ is positive definite $∀J∈𝒥$ and all of the principal sub-matrices of $_J$ are also positive definite if $ℐ = ∅$. Hence for any $≠̆$ we can find an $ε>0$ from \eqref{prop_eq:7_2} such that it satisfies \eqref{prop_eq:7_1}.
If $ℐ≠∅$ and given that $|ℐ|≤n-1$, then there exists a $k ∈⋂_J∈ℐJ$ from Lemma \ref{lem:6}. Let $K [n]\{k}$ and $_K$ is positive definite as $K∈𝒥\ℐ$.\\
Let $H∈ℋ_nr$ then $|H∩K| = |H| - 1$ and $k∈H$. To see this, if $H⊂K$ then $_H$ is positive definite matrix and this would imply that $H ⊂ℋ_fr$. As $_H∩K$ is principal sub-matrix of the matrix of $_K$ therefore it is full rank matrix. Hence $𝗋𝖺𝗇𝗄(_H∩K) = |H∩K| = |H| -1$. Observe,
\begin{align*}
|H| - 1 = |H\cap K| \leq \mathsf{rank}(\M_{H\cap K}) \leq \mathsf{rank}(\M_{H}) \leq |H|-1.
%\mathsf{rank}(\M_{H}) &\geq \\
\end{align*}
It follows that $(_H)  = |H| -1 ∀H ∈ℋ_nr$. This further implies that $(_H)≽$ and $𝗋𝖺𝗇𝗄((_H)) =1$. Let $^k$ be the $k^th$ column in the matrix $$ then $_H^k$ is a column $_H$ as $k∈H$. We have,
\begin{align*}
\adj(\M_H)\M_H = \det(\M)\mathbb{I}_{n-1} = \bz, \text{ which implies }~~~~\adj(\M_H)\m_H^k = \bz.
\end{align*}
Therefore $=̆ ^k$ satisfies \eqref{prop_eq:7_3} and $ε= 12 min_H ∈ℋ_fr(_H)^̆⊤_H(_H)_̆H$.
\end{proof}
% \begin{prop}\label{prop:6}
% Let $\M\in \mathcal{S}^n$ and \(J_i = [n]\sm\{i\}\) and $\m^i$ is the $i^{th}$ row of $\M$.Then $\det(\M) = m_{ii}\det(\M_{J_i}) -\m^{iT}_{J_i} \adj(\M_{J_i})\m_{J_i}^i$.
% \end{prop}
% \begin{proof}
% Without loss of generality take $i = 1$. From Laplace's equation expanding along the \(i^{th}\) column,
% \begin{equation}\label{prop_eq:6_1}
%     \det(\M) = \Sigma_{j=1}^n -1^{1+j} m_{1j}\det(\M_{1j})
% \end{equation}
% One can obtain $\M_{1j}$ from $\M_{11}$ by performing two operations on $\M_{11}$.
% The first operation is to replace $(j-1)^{th}$ row with $\m^{(1)}$ call the matrix obtained from the first operation $\M'_{1j}$ and
% second operation is to bubble $(j-1)^{th}$ row to the top what this means is sequentially interchange rows $j-1$ with $j-2$, interchange $j-2$ with $j-3$ so on till we get $\m^1$ to the top .
% $\M'_{1j}$ is same as $\M_{11}$ except the $(j-1)^{th}$ row. This means that the $(j-1)^{th}$ column of $\adj(\M_{ii})$ and $\adj(\M_{1j})$ are the same.

% \begin{align*}
%     \M'_{1j}\adj(\M'_{1j}) = \det(\M'_{1j}) \mathbb{I}\\
% \end{align*}
% \begin{equation}\label{prop_eq:6_2}
%     \det(M'_{1j}) = \adj(\M_{11})^{(j-1)^{th} \text{ column }} \m_{J_1}^{(1)}
% \end{equation}
% The second operation is to bubble the $(j-1)^{th}$ row to the top which takes $j-1$ row operation changes. So this means to multiply $\det(\M'_{1j})$ with $-1^{1+j-1}$\\
% From \eqref{prop_eq:6_2}, we get the determinant of
% \begin{equation}\label{prop_eq:6_3}
%     \det(\M_{1j}) = (-1)^j \adj(\M_{11})^{(j-1)^{th} \text{ column }} \m_{J_1}^1
% \end{equation}

% Combining \eqref{prop_eq:6_1} and \eqref{prop_eq:6_3},
% \begin{align*}
%     \det(\M) &= m_{11}\det(\M_{J_1}) +\Sigma_{j=2}^n -1^{1+j}\m_{1j} \det(\M_{1j})\\
%     \det(\M) &= m_{11}\det(\M_{J_1}) + \Sigma_{j=2}^n -1^{1+j}\m_{1j} (-1^j) \adj(\M_{11})^{(j-1)^{th} \text{row}}\m^{(1)}_{J_1}\\
%     \det(\M) &= m_{11}\det(\M_{J_1})- \Sigma_{j=2}^n  \m_{J_1}^{1\top} \adj(\M_{11})\m^1_{J_1}
% \end{align*}
% \end{proof}

% \begin{lem}\label{lem:8}
% Let $\M \in \Sminors{n}{n-1}$ and $\mathsf{rank}(\M_J) = n-2 \,\,\,\forall J\in[n]$ where $|J|=n-1$ and $\det(\M) < 0$.
% \begin{enumerate}
%     \item Then $\M - \varepsilon\u\u^{\top} \not \in \Sminors{n}{n-1}$ for all non-zero $\u\in \R^n$ and $\varepsilon \in \R_{>0}$.
%     \item If there exists $\A,\B\in \Sminors{n}{n-1}$ such that $\M  = \A + \B$, then there exists n-linearly independent vectors $\{\ve^i\}_{i=1}^n\in \R^n$ such that
%     \begin{align*}
%         \M_{J_i}\ve_{J_i}^i &= \bz\\
%         \A_{J_i}\ve_{J_i}^i &= \bz\\
%         \B_{J_i}\ve_{J_i}^i &= \bz
%     \end{align*}
%     where $J_i = [n]\backslash\{i\}$.
% \end{enumerate}
% \end{lem}
% \begin{proof}
% % Note $ \M $ cannot possess any zero columns under the given hypothesis. If it possess any zero columns then it would contradict the fact that $ \det(\M)<0 $.
% % Let $\mathcal{J} = \{J\in[n]\,\mid\, |J| = n-1\}$. Let $J\in \mathcal{J}$ and $\M_{J}$ is corresponding principal minor and $\det(\M_J) = 0$.
% %  The \(rank(\adj(\M_J)) = 1\) and it is a psd matrix
% %  therefore it can be written as  $\adj(\M_J) = \ve_J\ve_J^{\top}$
% % % $|\mathcal{J}| = n$.\\
% % \begin{align*}
% %     \det(\M_J - \varepsilon \u_J\u_J^\top) &= \det(\M_J) -\varepsilon \u^\top_J \adj(\M_J)\u_J\\
% %     \det(\M_J - \varepsilon \u_J\u_J^\top) &= - \varepsilon \u^\top_J \adj(\M_J)\u_J\\
% %      \u^\top_J \adj(\M_J)\u_J &= 0\\
% %      \adj(\M_J) \u_J &= \mathbf{0}\\
% %      \adj(\M_J) &= \ve_J\ve_J^\top\\
% %      \ve_J^\top\u_J &= 0
% % \end{align*}
% \begin{enumerate}
%     \item Assume to the contrary that there exists a rank 1 matrix $\u\u^{\top}$ such that $\M - \varepsilon \u\u^{\top} \in \mathcal{S}_+^{n,n-1}$.
% Let $J_i = [n]\backslash\{i\}$. Given that all $(n-1)\times (n-1)$ principal sub-matrices are positive semidefinite and have rank $n-2$, therefore $\adj(\M_{J_i}) = \w^i\w^{i{\top}}$ for some $\w^i \in \R^{n-1}$ and $\w^i\neq \bz$.
% Consequently,
% \begin{align*}
%     (\w^i)^\top\u_{J_i} &= 0
% \end{align*}

% \begin{align*}
% \ve^i\define \begin{cases}
%     \w_j^i & \forall j \in J_i\\
%     0 & \text{otherwise}
% \end{cases}
% \end{align*}

% Let
% \begin{align*}
%     \V = \begin{pmatrix}
%     ({\ve}^{1})^\top\\
%     ({\ve}^{2})^\top\\
%     \vdots\\
%     ({\ve}^{n})^\top
%     \end{pmatrix}
% \end{align*}.

% As there exists a \(\u\neq \mathbf{0}\) such that \( \V\u = \mathbf{0} \), then $\mathsf{rank}(\V) = r$ where $r < n$. Let the independent rows of $\V$ be $i_1,i_2,\ldots,i_r$. As $r<n$, there exists a $k \in \displaystyle\bigcap_{j=1}^r J_{i_j}$. Let $\m^{k}$ be the $k^{th}$ column of the matrix $\M$. We know that $\adj(\M_{J_{i_j}}) \m^k_{J_{i_j}} = \bz$ since index $k \in J_{i_j} \forall j \,=\,1,2,\ldots,r$ and $\M_{J_{i_j}}$ is rank-deficient matrix. This implies that $ \m^{k} \in \text{nullspace}(\V)$ and hence $(\w^k)^\top\m^{k}_{J_k} = 0\implies \adj(\M_{J_k}) \m_{J_k}^{k} = \bz$.

% But from Lemma \ref{lem:4}, we have that $ \det(\M) = - (\m^k_{J_k})^{\top}\adj(\M_{J_k}) \m_{J_k}^{k}$ and $\det(\M)<0$ which contradicts the given assumption. Hence there exists no rank one matrix $\u \in \mathbb{R}^{n}$.
% \item Let $\ve^i$ be the vectors defined in the earlier proof. Observe that
% \end{enumerate}

% % Therefore \(\V\) is full-rank matrix and hence there exists no rank-1 matrix.
% \end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%PROPOSITION 9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{prop}\label{prop:9}
Let $\M\in \Sminors{n}{n-1}$ and $\mathsf{rank}(\M_{J}) = n-2\,\,\forall J\in [n]$ where $|J| = n-1$ and $\det(\M) < 0$. If there exists $\A,\B\in \Sminors{n}{n-1}$ such that $\M = \A + \B$, then there exists $n$ linearly independent vectors $\{\ve^i\}_{i=1}^n$ such that
\begin{align*}
\M_{J_i}\ve_{J_i}^i &=\bz\\
\A_{J_i}\ve_{J_i}^i &= \bz\\
\B_{J_i}\ve_{J_i}^i &= \bz
\end{align*}
where $J_i = [n]\backslash\{i\}$ and $i\in[n]$. Furthermore, all the diagonal elements of $\M$ are non-zero.
\end{prop}
\begin{proof}
From Lemma \ref{lem:5}, we have that $\adj(\M_{J_i}) =\w^i{\w^i}^{\top}$ where $J_i\define[n]\backslash\{i\}$ and $i\in[n]$.
\begin{align*}
\M_{J_i}\adj(\M_{J_i}) &= \det(\M_{J_i})\mathbb{I}\\
\M_{J_i}\w^i{\w^i}^\top &= \bz\\
(\w^i)^{\top} \M_{J_i}\w^i &= 0\\
\M_{J_i}\w^i &= \bz\,\,\,\,(\text{As $\M_{J_i}$ positive semidefinite.)}
\end{align*}
\begin{align*}
\ve^i\define\begin{cases}
\w_j^i& \text{  for } j \in J_i\\
0 & \text{otherwise}
\end{cases}
\end{align*}
Observe that $\ve_{J_i}^i = \w^i$. As $\M = \A + \B$, we have that
\begin{align*}
\M_{J_i} &= \A_{J_i} + \B_{J_i}\\
(\w^i)^{\top}\M_{J_i}\w^i &= (\w^i)^{\top}\A_{J_i}\w^i + (\w^i)^{\top}\B_{J_i}\w^i\\
0 &=(\w^i)^{\top}\A_{J_i}\w^i + (\w^i)^{\top}\B_{J_i}\w^i\\
\end{align*}
As $\A_{J_i},\B_{J_i} \in \mathcal{S}_+^{n-1}$, we have that
\begin{align*}
(\w^i)^{\top}\A_{J_i}\w^i &=  (\w^i)^{\top}\B_{J_i}\w^i = 0\\
\A_{J_i}\w^i &=  \B_{J_i}\w^i= \bz
\end{align*}
We now show that $\{\ve^i\}_{i=1}^n$ is linearly independent. Assume to the contrary that $\{\ve^i\}_{i=1}^n$ are linearly dependent. Let
\begin{align*}
\V = \begin{pmatrix}
\vdots\\
\end{pmatrix}.
\end{align*}
There exists a $\u\in\R^n$ such that $\V\u =\bz$, this implies that $\mathsf{rank}(\V) = r<n$. Let the independent rows of $\V$ be $i_1,i_2,\ldots,i_r$. As $r<n$, there exists a $k \in \displaystyle\bigcap_{j=1}^r J_{i_j}$. Let $\m^{k}$ be the $k^{th}$ column of the matrix $\M$. We know that $\adj(\M_{J_{i_j}}) \m^k_{J_{i_j}} = \bz$ since index $k \in J_{i_j} \forall j \,=\,1,2,\ldots,r$ and $\M_{J_{i_j}}$ is rank-deficient matrix. This implies that $ \m^{k} \in \text{nullspace}(\V)$ and hence $(\w^k)^\top\m^{k}_{J_k} = 0\implies \adj(\M_{J_k}) \m_{J_k}^{k} = \bz$.

But from Lemma \ref{lem:4}, we have that $ \det(\M) = - (\m^k_{J_k})^{\top}\adj(\M_{J_k}) \m_{J_k}^{k}$ and $\det(\M)<0$ which contradicts the given assumption. Hence $\V$ is a full rank matrix.

Assume to the contrary that $\M$ has at least one diagonal element which is zero. Let this element be $m_{ii} = 0$. As $m_{ii}=0$ this implies that $m_{ik} = 0$ to see this
\begin{align*}
m_{ii}m_{kk} -m_{ik}^2 &\geq 0\,\, \because\,\, \M\in \Sminors{n}{n-1}\\
m_{ik}^2 &= 0.
\end{align*}
This means that if $\M$ has a zero column, which is the $i^{th}$ column, then $\det(\M) = 0$ which contradicts the assumption that $\det(\M) <0$. \ignore{Hence proved.}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%PROPOSITION 9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%COROLLARY 9 9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{cor}\label{cor:9}
% {Let $\M \in \Sminors{n}{n-1}$ and rank of all $(n-1)\times (n-1)$ principal minors is $n-2$ and $\det(\M) < 0$.
% If $\M = \A + \B$ for some $\A,\B \in \Sminors{n}{n-1}$ then neither $\A$ nor $\B$ belongs to $\mathcal{S}_{+}^n$.}
% \end{cor}
% \begin{proof}
% Proof by Contradiction,\\
% Without loss of generality assume that $\A \in S'$ this means that one of the principal minors of $\A$ has at most rank $n-3$. From Proposition \ref{prop:6}, $\A$ is positive semidefinite matrix.
% Given $\M = \A + \B$ and as $\A$ is positive semidefinite  \(\A = \Sigma_{j=1}^n \lambda_j \mathbf{v}_j\mathbf{v}_j^{\top}\) and $\lambda_j \geq 0$.
% \begin{align*}
%     \M &= \A + \B\\
%     \M &= \Sigma_{j=1}^n \lambda_j \mathbf{v}_j\mathbf{v}_j^{\top} + \B
% \end{align*}
% \begin{equation}
%         \M - \lambda_1 \mathbf{v}_1\mathbf{v}_1^{\top} = \Sigma_{j=2}^n \lambda_j\mathbf{v}_j\mathbf{v}_j^{\top} + \B
% \end{equation}
% The right-hand side of the above equation belongs to $\Sminors{n}{n-1}$ this implies there exists a rank-1 matrix such that $\M - \lambda_1 \mathbf{v}_1\mathbf{v}_1^{\top} \in \Sminors{n}{n-1}$.
% But this is impossible due to Lemma [\ref{lem:8}].
% Hence a contradiction.
% \end{proof}
% \begin{lem}\label{lem:10}
% Let $\M \in \Sminors{n}{n-1}$ and rank of all $(n-1)\times (n-1)$ principal sub-matrices is $n-2$
% and $\det(\M) < 0$.
% Let $S' = \{\A \in \Sminors{n}{n-1}|\A\text{ has at least one $(n-1)\times (n-1)$ principal sub-matrix with rank } n-1 \}$.
% If $\M = \A + \B$ then neither $\A$ nor $\B$ belongs to $S'$.
% \end{lem}
% \begin{proof}
% Given $\M$ has all of its $(n-1) \times (n-1)$ principal sub-matrices of rank $n-2$ and $\M = \A + \B$. Assume to the contrary and without loss in generality let $\A \in S'$. Let the principal minor of $\A$ whose rank is $n-1$ let it be $\A_{11}$.

% We know that $\M_{11}\in \mathcal{S}_{+}^{n-1}$ is a rank $n-2$ that means there exists a vector $\u \in \mathbb{R}^{n-1}$ such that $\u^\top\M_{11}\u = 0$ and $\u^\top\A_{11}\u \not = 0$.
% \begin{align*}
%     \u^\top\B_{11}\u \geq 0\\
%     \u^\top\A_{11}\u > 0\\
%     \u^\top\A_{11} + \B_{11}\u >0\\
%     \u^\top \M_{11} \u > 0\\
%     0> 0
% \end{align*}
% This is a contradiction. Hence neither $\A$ nor $\B$ belong to $S'$.
% \end{proof}
\begin{lem}\label{lem:11}
Let $\M \in \Sminors{n}{n-1}$ and $\det(\M)<0$. Let $G_{\M}(V,E)$ be a graph associated with the matrix $\M$ with \ignore{which is defined as follows} $V = [n]$ and $ij \in E$ if $m_{ij} \neq 0$. Then $G_{\M}(V,E)$ is a connected graph.
\end{lem}
\begin{proof}
Assume to the contrary that $G_{\M}(V,E)$ is disconnected. Then $V = \bigcup_{i=1}^k V_i$  where the induced subgraph on \(V_i\) is connected and $1\leq |V_i| \leq n-1$ and \(V_i\cap V_j = \emptyset\) for \(i\neq j\).
We can partition the matrix $\M$ into the following form,
\begin{align*}
\M =  \begin{pmatrix}
\M_{V_1} & \mathbf{0} & \ldots &\mathbf{0}\\
\mathbf{0} & \M_{V_2} & \ldots & \mathbf{0} \\
\ldots & \ldots & \ldots & \ldots \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \M_{V_k}
\end{pmatrix}.
\end{align*}
Each $\M_{V_k}$ is a positive semidefinite since $\M_{V_k}$ is a principal sub-matrix of size less than or equal to $n-1$. For any $\x\in \mathbb{R}^n$,\\
\begin{align*}
\x^\top\M\x &= \Sigma_{i=1}^k \x^\top_{V_{i}}\M_{V_{i}}\x_{V_{i}}\geq 0.
\ignore{\x^\top\M\x &\geq 0.}
\end{align*}
From this we can conclude that $\det(\M)\geq 0$. But this violates the given condition that $\det(\M) < 0$. Hence $G_{\M}(V,E)$ is a connected graph.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%PROOF OF THEOREM 5.3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Proof of Theorem \ref{prop:extrays3}}\label{proof_thm5.3}
\begin{proof}
Let\ignore{'s assume to the contrary that $\M$ does not span an extreme ray then one can write} $\M =  \A + \B$ where $\A,\B\in\Sminors{n}{n-1}$. We will show that $\A,\B$ are positive scalar multiples of $\M$.
\ignore{Now, it follows from Corollary \ref{cor:9} and Lemma \ref{lem:10} that $(n-1)\times (n-1)$ principal sub-matrices of both $\A$ and $\B$  are of rank $n-2$ and $\det(\A),~\det(\B)<0$.}
Let $J_i = [n]\sm\{i\}$. From Proposition \ref{prop:9}, there exists linearly independent vectors $\{\ve^i\}_{i=1}^n$ such that
\begin{equation*}
\M_{J_i}\ve_{J_i}^i = \A_{J_i}\ve_{J_i}^i = \B_{J_i}\ve_{J_i}^i = \bz.
\end{equation*}
% \begin{align*}
%     \M & = \A + \B\\
%     \M_{J_i} & =  \A_{J_i} + \B_{J_i}\\
%     \M_{J_i}\adj(\M_{J_i}) & =  \A_{J_i}\adj(\M_{J_i}) + \B_{J_i}\adj(\M_{J_i})\\
%     \adj(\M_{J_i})& = \w^i(\w^i)^{\top}\\
%     \bz & = \A_{J_i} \w^i(\w^i)^{\top} + \B_{J_i} \w^i(\w^i)^{\top}
% \end{align*}

% \begin{align*}
% \ve^i\define \begin{cases}
%     \w_j^i & \forall j \in J_i\\
%     0 & \text{otherwise}
% \end{cases}
% \end{align*}
% \begin{align*}
%     \V  = \begin{pmatrix}
%     (\ve^1)^\top\\
%     (\ve^2)^\top\\
%     \vdots\\
%     (\ve^n)^\top\\
%     \end{pmatrix}
% \end{align*}
% \begin{align*}
%     \A_{J_i}\w^i & = \mathbf{0} \\
%     \B_{J_i}\w^i & = \mathbf{0} \\
%     \M_{J_i}\w^i & = \mathbf{0}
% \end{align*}
Now, $S \define \Bigl\{\X \in \mathcal{S}^n:\X_{J_i} \ve_{J_i}^i = \mathbf{0} \,\,\,\forall\,\,\, J_i\in \mathcal{J}\Bigl\}$. $S$ is a subspace of the vector sp space of symmetric matrices, $\mathcal{S}^n$. The number of equations that define $S$ are $n(n-1)$ and $\X$ has a total of $\dfrac{n(n-1)}{2}$ variables.
Hence $\mathsf{dim}(S)\leq \dfrac{n(n-1)}{2}$. Observe that $\M,\A,\B \in S$.\\

Let $i\in[n]$ and $\x^i$ be a column of $\X$, then we claim that,
\begin{align}\label{prop:extrays3:eq1}
(\ve^j)^{\top}\x^i =\bz \,\,\,\,\,\forall j\in J_i.
\end{align}

To see this,
\begin{align*}
\mathbf{v}^{j}_{J_j} \mathbf{x}^{(i)}_{J_j}  = 0,\\
\sum_{k\in J_j} v_k^jx_k^i = 0
\end{align*}

As $v_j^j= 0 $ from Proposition \ref{prop:9}, we have
\begin{align*}
\sum_{k\in J_j} v_k^jx_k^i + v_j^jx_j^i = 0 = (\mathbf{v}^{j})^{\top}\x^i
\end{align*}
and \eqref{prop:extrays3:eq1} holds.
\ignore{The last but on equality follows from the definition of $ \ve^{j} $ [Proposition \ref{prop:9}] where $ v_j^j= 0 $.}
Let $\V$ be the matrix defined in \ref{prop:9} and the rows of the matrix are $ \{\ve^i\}_{i=1}^n $. Let $\mathbf{V}(J_i)$ be the rows of the matrix $\V$ indexed by $J_i$.
Let $\mathbf{x}^{i}$ be the $i^{th}$ row of the matrix of $\X$. It follows from \eqref{prop:extrays3:eq1}, that
\begin{align*}
\V(J_i)\x^i = 0.
\end{align*}
\ignore{From Proposition \ref{prop:9}, we know that $\V$ is a full rank and hence $\V(J_i)$ is a $(n-1) \times n$ full-rank matrix. But $\V(J_i)$ has $n-1$ rows and $n$ columns.} It follows from Proposition \ref{prop:9} that $\V(J_i)$ is an $(n-1) \times n$ full-rank matrix.
Select all the independent columns of $V(J_i)$ and the $ (n-1)\times(n-1)$ matrix formed by these columns is an invertible square matrix. \ignore{The leftover column let it be the $k^{th}$ column.} Denote by $k$, the column which is not part of the above invertible $(n-1)\times(n-1)$ matrix. Hence, we can express every $x_{ij}$ in terms of $x_{ik}$.

Every $x_{ij}$ of the $\X$ is scalar multiple $x_{ik}$.
The diagonal elements of the matrix $ \X $ are non-zero, because if at least one of them is zero, let this element be the $ x_{ll} $ where $ l\in [n] $, then we would have that $ m_{ll} = 0$  this impossible from Proposition \ref{prop:9}. Therefore,
\begin{align*}
x_{qq} &\not =  0\,\,\forall q\in[n],\\
x_{ii} &= c_{ik-ii}x_{ik},
\end{align*}
which implies that $c_{ik-ii} \neq 0$. This further suggests that every $x_{ij}$  is scalar multiple of $x_{ii}$. Similarly, one can obtain a relation between $x_{kk}$ and $x_{ik}$. Hence,
\begin{align*}
x_{kk} &= c_{ik-kk} x_{ik}\\
&= \dfrac{c_{ik-kk}}{c_{ik-ii}}x_{ii}\\
&= C_{ii-kk}x_{ii},
\end{align*}
where $ C_{ii-kk} = \dfrac{c_{ik-kk}}{c_{ik-kk}}$.

From the elimination of variables process let $P$ be the set of indices such that $x_{ij} = 0$ regardless of the matrix in $S$. Let $G_{\X}(V,E)$ \ignore{where $V = [n]$ and $ij \in E$ if $ij \not \in P$} be the graph of $\X$ as defined in Lemma \ref{lem:11}. From Lemma $\ref{lem:11}$ we know that $G_{\M}(V,E)$\ignore{, where $V = [n]$ and $ij \in E$ if $m_{ij} \neq 0$,} is a connected graph \ignore{$G_{\M}$} and is a subgraph of $G_{\X}$. \ignore{Hence} Therefore $G_{\X}$ is a connected graph.

For each edge $pq$ of the graph $G_{\X}$ one can establish a relation between $p$ and $q$. As $G_{\X}$ is connected, one can find a path between any two nodes in the graph. For example, there is a path from $i$ to $i'$ let this path be $i - p_1 - \ldots - p_l - i'$. It follows that,
\begin{align*}
x_{p_1p_1} &= C_{ii-{p_1p_1}}x_{ii}\\
x_{p_2p_2} &= C_{{p_1p_1}-{p_2p_2}}x_{p_1p_1}\\
x_{i'i'} &= C_{{p_lp_l}-{i'i'}}x_{p_lp_l}.
\end{align*}
This implies $x_{i'i'}$ is a scalar multiple of $x_{ii}$. The constants $c's$ and $ C's $ are obtained from the sub-matrix inverse of $\V$ and $\V$ is obtained from $\M$ and do not depend on $\X$. \ignore{Hence $c$'s and $ C $'s are constants regardless of $\X$.}
Hence the rank of the system $S$ is 1 and every matrix in $S$ can be written as $a\mathbf{M}'$ where $a\in \R$. Since $\A,\B,\M \in \mathcal{S}_+^{n,n-1}$, we can write $\A = a_{\A}\mathbf{M}'$, $\B = a_{\B}\mathbf{M}'$, and $\C = a_{\C}\mathbf{M}'$ where $a_{\A},a_{\B},a_{\M}\in \R$. Therefore $\A = \dfrac{a_{\A}}{a_{\M}} \M$ and $\B = \dfrac{a_\B}{a_{\M}} \M$ . Hence $\M$ spans an extreme ray of $\Sminors{n}{n-1}$.
\end{proof}

\begin{cor}\label{cor:12}
For $n\geq 3$,\ignore{define} \(G(n,n-1) = \dfrac{n-1}{n-2} \mathbb{I}_{n} - \dfrac{1}{n-   2}\mathbf{1}_{n}\mathbf{1}_{n}^\top\)\ignore{.Then $G(n,n-1)$} spans an extreme ray of \(\Sminors{n}{n-1}\)
\end{cor}
\begin{proof}
It suffices to prove that \(\det(G(n,n-1)) < 0\) and all \((n-1)\times (n-1)\) principal sub-matrices of $G(n,n-1)$ are positive semidefinite and have rank \(n-2\) (Theorem \ref{prop:extrays3}).

Observe that the eigenvalues of $G(n,n-1)$ are $\frac{-1}{n-2}$ with multiplicity 1, and $\frac{n-1}{n-2}$ with multiplicity $n-1$. We have $\det(G(n,n-1)) = -\frac{1}{n-2}\left(\frac{n-1}{n-2}\right)^{n-1} < 0.$ Furthermore, every $(n-1)\times (n-1)$ principle sub-matrix of $G(n,n-1)$ can be expressed as \(\frac{n-1}{n-2}\mathbb{I}_{n-1} - \frac{1}{n-2}\mathbf{1}_{n-1}\mathbf{1}^{\top}_{n-1} \). The eigenvalues of every  $(n-1)\times (n-1)$ principle sub-matrix of $G(n,n-1)$ are thus $0$ with multiplicity 1, and $\frac{n-1}{n-2}$ with multiplicity $(n-2)$. Consequently, every $(n-1)\times (n-1)$ principle sub-matrix of $G(n,n-1)$ is positive semidefinite with rank $(n-2)$.
\ignore{As discussed earlier, the corollary follows from Theorem \ref{prop:extrays3}. However, to further illustrate the proof technique of Theorem \ref{prop:extrays3}, in the following, we utilize the same tools to prove that $G(n,n-1)$ spans an extreme ray of $\Sminors{n}{n-1}$.

Assume to the contrary, \(G(n,n-1)\) does not span an extreme ray of \(\mathbb{S}_+^{n,n-1}\)
and \(G(n,n-1) = \A + \B\) where \(\A,\B \in \mathcal{S}_+^{n,n-1}\). Let \(J \subset [n]\) and \(|J| = n-1\). We have,
\begin{align*}
G(n,n-1)_J &= \A_J + \B_J
\end{align*}
Multiplying on left and right by $\mathbf{1}_{n-1}$,
\begin{align*}
0=\mathbf{1}_{n-1}^{\top}\bigg(\frac{n-1}{n-2}\mathbb{I}_{n-1} - \frac{1}{n-1}\mathbf{1}_{n-1}\mathbf{1}_{n-1}^{\top}\bigg)\mathbf{1}_{n-1} &= \mathbf{1}_{n-1}^{\top}\A_J\mathbf{1}_{n-1} + \mathbf{1}_{n-1}^{\top}B_J\mathbf{1}_{n-1}
\ignore{0 &= \mathbf{1}_{n-1}^{\top}\A_J\mathbf{1}_{n-1} + \mathbf{1}_{n-1}^{\top}\B_J\mathbf{1}_{n-1}.}
\end{align*}
It follows that $\mathbf{1}_{n-1}^{\top}\A_J\mathbf{1}_{n-1} = 0$ and $\mathbf{1}_{n-1}^{\top}\B_J\mathbf{1}_{n-1} = 0$. Alternatively, $\A_J\mathbf{1}_{n-1} = \mathbf{0}$ and $\B_J\mathbf{1}_{n-1} = \mathbf{0}$.

Let \(J_i = [n]\sm[i]\). Select all the equations indexed by index \(i\) in \(\A_{J_{j}}\mathbf{1} = 0\) where \(j\in J_i\).
The equations are of the form,
\begin{align*}
\Sigma_{k\in J_j}a_{ik} = 0\,\, \forall j \in J_i.
\end{align*}
Let \(l_1,l_2\in J_i\) then \(|J_{l_1} \cap J_{l_2}| = n-2\).
The above equations can be rewritten as,
\begin{align*}
a_{ii} + a_{il_2} + \Sigma_{k\in J_{l_1}k\neq i,l_2} a_{ik} = 0,\\
a_{ii} + a_{il_1} + \Sigma_{k\in J_{i_2}k\neq i,l_1} a_{ik} = 0,\\
a_{il_2} = a_{il_1}.
\end{align*}
This implies \(a_{ik} = a_{il_1}\forall k\in J_{i}\). Therefore \ignore{we have that},
\begin{align*}
a_{ii} + \Sigma_{k\in J_{l_2},k\neq i} a_{ik} &= 0,\\
a_{ik} &= a_{il_1} \,\, \forall k\in J_i,\\
a_{ii} + (n-1) a_{il_1} &= 0.
\end{align*}
Simplifying, we obtain
\begin{align*}
a_{il_1} &= \frac{-1}{n-1}a_{ii}, \text{ and }\\
a_{ik} &= \frac{-1}{n-1}a_{ii} \forall k \in J_i.
\end{align*}

Select all the equations indexed by \(k\) in \(A_{J_j}\mathbf{1} = 0\) where \(j\in J_k\).
Repeating the above procedure \(a_{kl} = \frac{-1}{n-1}a_{kk}\) for all \(l \in J_{k}\). Hence
\begin{align*}
a_{ik} &= a_{ki}\\
a_{ik} &= \frac{-1}{n-1}a_{ii}\\
a_{ki} &= \frac{-1}{n-1}a_{kk}\\
a_{ii} &= a_{kk} ~\forall i\neq k.
\end{align*}
This implies that \(\A = a_{11}G(n,n-1)\) and \(B = b_{11}G(n,n-1)\).
But this is a contradiction. Hence \(G(n,n-1)\) spans an extreme ray of \(\Sminors{n}{n-1}\).}
\end{proof}

\Gnkextrays*
\begin{proof}
Assume that \(G(n,k)\) does not span an extreme ray of \(\Sminors{n}{k}\). Then \(G(n,k) = \A + \B\) where \(\A,\B \in \Sminors{n}{k}\).
Let \(J\in[n]\) and \(|J| = k+1\). It follows,
\begin{equation*}
(G(n,k))_J = G(k+1,k)
\end{equation*}
\begin{align*}
G(n,k)_J &= \A_J + \B_J\\
G(k+1,k) &= \A_J + \B_J\\
j &= \min(J)\\
\A_J &= a_{jj}G(k+1,k)\\
\B_J &= b_{jj}G(k+1,k).
\end{align*}
This implies \(\A = a_{11} G(n,k)\) and \(\B = b_{11}G(n,k)\),\ignore{This is a contradiction. It follows that}and \(G(n,k)\) spans an extreme ray of \(\Sminors{n}{k}\).
\end{proof}
\NLSextremerays*
% \begin{cor}\label{cor:13}
%     \(G(n,k) = \dfrac{k}{k-1}\mathbb{I}_n - \dfrac{1}{k-1}\mathbf{1}_n\mathbf{1}_n^{\top}\) spans an extreme ray of \(\Sminors{n}{k}\).
% \end{cor}
\begin{proof}

We know that $\M$ is diagonally congruent to $G(n,k)$ [Blekherman et~al., 2022]. Assume to the contrary that $\M$ doesn't span an extreme ray then we have that $\M = \M_1 + \M_2$ where $\M_1,\M_2\in \calS^{n,k}_+$ and $\M_1 \neq \M_2$.
Since $\M$ is diagonally congruent to $G(n,k)$ we have that there exists an invertible diagonal matrix such that $\M = \mathbf{D}G(n,k)\mathbf{D}$. It follows,
\begin{align*}
\M &= \M_1 + \M_2\\
\mathbf{D}^{-1}\M\mathbf{D}^{-1} &= \mathbf{D}^{-1}\M_1\mathbf{D}^{-1} + \mathbf{D}^{-1}\M_2\mathbf{D}^{-1}.
\end{align*}
From Sylvester's law of inertia and as $\mathbf{D}^{-1}$ is a diagonal matrix we have that $\mathbf{D}^{-1}\M_1\mathbf{D}^{-1},\mathbf{D}^{-1}\M_2\mathbf{D}^{-1} \in \calS^{n,k}_+$. This contradicts corollary \ref{cor:Gnkextrays}. Hence $\M$ spans an extreme ray of $\Sminors{n}{k}$.

% \end{proof}

% % A matrix $\M\in \calS^{n,k}_+$ is called non-singular locally singular (\emph{NLS}) matrix for $n\geq 5$ and $3 \leq k\leq n-2$ if all the $k\times k$ principal minors are zero and $\det(\M)\neq 0$ [Blekherman et~al., 2022].  The set of NLS matrices in $\calS^{n,k}_+$ are \textit{diagonally congruent} to $G(n,k)$ for all $n\geq 5$ and $3\leq k\leq n-2$.
% \begin{cor}\label{cor:14}
%     An $NLS$ matrix $\M \in \calS^{n,k}_+$, for $n\geq 5$ and $3\leq k\leq n-2$, spans an extreme ray of $\calS^{n,k}_+$.
% \end{cor}
% \begin{proof}

\end{proof}
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