Questions tagged [matrix-analysis]
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
768 questions
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When do all extreme points of a set of matrices have their extreme matrix norm equal to the joint spectral radius?
Let $E$ be a compact convex set of endomorphisms of $\mathbf{R}^n$ and $\varphi$ be the associated Barabanov norm; cf. Definition 2.3 in [1]. Let $\rho(E)$ be the joint spectral radius of E; cf. ...
6
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2
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488
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A question related to matrix inverse diagonal zero property
$\DeclareMathOperator\supp{supp}$Let a symmetric nonsingular matrix $A \in \mathbb{R}^{2n \times 2n} $ have the following block form
$$ A = \begin{bmatrix}
X & D \\
D^{\top} ...
- 61
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1
answer
85
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perturbation of k-sum of eigenvalues
Let $A$ be an $n\times n$ symmetric matrix and $S$ be an $n\times n$ positive definite matrix.
Suppose the eigenvalues of $A$ and $SAS$ are arranged in non decreasing order; that is
$$
\lambda_1 (A) \...
- 39
1
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1
answer
169
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How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
- 75
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424
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A question in matrix analysis and linear algebra related to Hermitian Toeplitz matrix
Consider a Hermitian Toeplitz matrix and modify it with entries in leading diagonal as $H_{n\times n}(x,x) = x^2 + \lambda,x=0,1,2,\dotsc n-1$. Now we choose $\lambda\in\mathbb{R}$ such that the ...
- 544
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0
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99
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Variant of Cordes Inequality
The classical Cordes inequality states the following: suppose that $\|\cdot\|$ is the usual matrix norm, $0 < \alpha \leq 1$, and $A, B$ are $n \times n$ positive semidefinite Hermitian matrices. ...
19
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2
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1k
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Off-diagonal entries of $(AA^T)^{1/2}$ for the bipartite adjacency matrix of a tree
Let $T$ be a tree with a bipartition of its vertices into sets $X = \{v_1, \dots, v_m\}$ and $Y = \{w_1, \dots, w_n\}$. Define the $m \times n$ bipartite adjacency matrix $A$ by
$$ A_{ij}=\begin{cases}...
- 4,503
1
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0
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145
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Error characterization for $([C]_S)^{\alpha}-[C^{\alpha}]_S$ in SPD Matrices
Let $S\subset \left\{1,2,\ldots,N\right\}$ and let $[C]_{S}\in\mathbb{S}^{|S|}$ be the principal submatrix of $C$ indexed by $S$. We may view $[\cdot]_S\,:\,\mathbb{S}^{N}\rightarrow \mathbb{S}^{|S|}$ ...
- 787
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93
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Closure of the Karpelevič region
In 1938, Kolmogorov posed the problem of characterizing the set
$$\Theta_n := \{ \lambda \in \mathbb{C} \mid \lambda \in \sigma(A),\ A \geqslant 0,\ Ae = e\},$$
where $e$ denotes the all-ones vector.
...
- 1,943
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153
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Solving an SDP that involves a decreasing sequence of circulant and skew-circulant matrices
Question. I would like to find an analytic solution of the following semidefinite program, where $e_0 = (1, 0, \ldots, 0)^{\top}$.
$$
\begin{aligned}
y = \min &\frac{1}{n} \sum_{i,j=0}^{n-1} A[i,j]...
- 81
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0
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131
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Lipschitz property of Frobenius norm of "standard deviation matrix"
Question: Is the Frobenius norm of (some form of) standard deviation matrix Lipschitz with respect to the Wasserstein distance?
To be more precise: Suppose $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ are two-...
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123
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Characterization of positive definiteness of product of two positive definite matrices with special structures
Let $A$ be $n\times n$ matrix with special structure $A:=I_n+a\cdot \mathbf{1}\mathbf{1}^{\rm T}$, where $I_n$ is $n\times n$ identity matrix, $a>0$ is a scalar and $\mathbf{1}$ is an $n$-...
- 327
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165
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Meta-principle for approximation of kernels
Suppose, we have an approximation result for finite matrices of all orders $n \times n$. When can we push such a result in the case of infinite matrices or kernels, maybe via possibly some ultralimit ...
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470
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A prototypical problem for transfer matrix calculations in combinatorics
Let $G$ be a directed graph with self-loops on the positive integers: for every $n$, $G$ has the directed edges $(n,n)$ and $(n, n+1)$; additionally, if $n$ is even, $G$ has the directed edge $(n, n/2)...
- 9,741
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111
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A reverse choice of a norm on a finite-dimensional Banach module
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$In the following, everything is finite-dimensional over a base field $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$.
Let $(A,\norm{\cdot}_A)$ be an ...
- 7,933
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128
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Is there an exact formula for the geometric mean of two complex semi definite matrices?
I found it interesting that there is a concept of geometric mean for positive definite matrices. At first I wanted to see if it was possible to extend it to semi definite matrices which I was able to ...
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197
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Is the complete polarization of the determinant non-negative on tuples of hermitian positive semidefinite matrices?
$\DeclareMathOperator{\tr}{tr}$
I will explain my question for $n = 3$. Assume first that $A$ is a complex $3 \times 3$ matrix. If the eigenvalues of $A$ are $\lambda_i$, for $i = 1, \dots, 3$, we ...
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235
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Bounds on the largest singular value computable in $O(n^2)$
Consider an $n \times n$ matrix $A$. I'm interested in algorithms that can verify whether the largest singular value of $A$, i.e., its spectral norm $\| A \|_2$, is less than or equal to $1$ or not. ...
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118
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Determinant of a exponential/vandermonde type matrix
Let $x,y\in\mathbb{R}^n_{+}$, and define $A=[a_{ij}]_{i,j=1}^n\in\mathbb{R}^{n\times n}$ such that $a_{ij}=e^{x_iy_j}$. Do we know a closed form for $\text{det}(A)$? I'm interested in this because of ...
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On the reducibility of a characteristic polynomial
Given $n \times n$ Hermitian matrices $A$ and $B$, we define the polynomial,
$$g(\alpha, \beta) := \det(A + \alpha B - \beta I),$$
where $I$ is the $n \times n$ identity matrix. It may be seen as the ...
8
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1
answer
336
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Is a smooth function of an operator smooth?
Let $f\colon \mathbb R\to\mathbb R$ be a function and $A\in M_n:=Hom(\mathbb R^n,\mathbb R^n)$ a linear operator with real eigenvalues and diagonal Jordan form. Then one naturally defines an operator $...
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Decomposition of a matrix relative to a diagonal matrix: how to prove $\tilde M=\tilde \Omega +\tilde B +\tilde N \Lambda^{-1}$?
Let $\Lambda$ be a non-degenerate $n \times n$ diagonal matrix with distinct non-zero entries. It is known (see Constitutive laws for the matrix-logarithm of the conformation tensor by Fattal and ...
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How to construct a matrix ($n\times m$, $m>n$) over $\mathbb{F}_2 $ with full rank, with the row width $m$ and entries' weight per row minimized?
How to construct a full-rank $n \times m$ matrix over $\mathbb{F}_2$ with $m > n$, minimizing width and row sparsity?
Goal:
Construct a matrix $X \in \mathbb{F}_2^{n \times m}$, with $m > n$, ...
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Is partial trace operator Lipschitz?
$\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\tr{tr}$Let $\mathcal H$ denote a Hilbert space, either $\mathcal H=\mathbb C^d$ ($d$-dimensional complex space), either $\mathcal H=L^2(\mathbb R^d, \...
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Positive definiteness of negative exponential of a matrix-valued quadratic form
For a matrix $M$ denote by $M^*$ its Hermitian conjugate.
For integers $d,m \geq 1$ consider the function $f:\mathbb{R}^m \to \mathbb{C}^{d \times d}$ defined for a column vector $\boldsymbol{x}=(x_1,\...
- 309
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1
answer
312
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Positive definiteness of some matrix-valued exponential kernel
For a matrix $M$, let $M^*$ denote the Hermitian conjugate of $M$. Fix integers $d \geq 1$ and $m \geq 1$. Consider the matrix-valued function $f : {\Bbb C}^{d \times m} \to {\Bbb C}^{d \times d}$ be ...
- 309
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Non-negativity of a matrix initial value problem
This is a follow-up to this. For $t \in [0,1]$, let $A(t)$ be a time-varying symmetric $n \times n$ matrix which is twice differentiable w.r.t. $t$. Let $X(t)$ be a time-varying $n \times n$ matrix ...
- 409
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176
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Estimation for solution of the Lyapunov equation with semidefinite right-hand side
Consider the Lyapunov matrix equation in symmetric matrix unknown $\bf X$
$$ {\bf A}^\top {\bf X} + {\bf X} {\bf A} = − {\bf B} {\bf B}^\top$$
where the matrix $\bf A$ is Hurwitz. We know that its ...
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Bounds on the eigenvalues of perturbations of a symmetric matrix
This question was previously posted on MSE.
Let us fix $\varepsilon\in (0,1)$ and $\beta\in\mathbb R$.
Consider the $2 n\times 2n$ symmetric tridiagonal probability matrix
$$Q_n :=\begin{bmatrix}
1-\...
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0
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Matrix norm of a deterministic diagonal operator with sparse random gaussian non-diagonal entries
I am considering the usual matrix norm i.e. $||A||=\lambda_{max}(A)$ since our matrices are all symmetric.
We let $N$ be an integer and let $X_1,\dots X_{K_N}$ be a set of i.i.d centered gaussian ...
- 1
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3
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Almost every Hermitian matrix has distinct eigenvalue differences
I know that almost every Hermitian $n \times n$ matrix $A$ has distinct eigenvalues $\lambda_1$, $\ldots$, $\lambda_n$. (There's a proof in the "spectral dynamics" section of this blog post.)...
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Singular values and eigenvalues of matrices with coefficients in an ultrametric field
Consider a complete ultrametric field $(\Omega,|.|)$. We endow $M_n(\Omega)$ with the maximum norm. Given a matrix $G\in M_n(\Omega)$, recall that the singular values $\sigma_1\geq\dots \geq \sigma_n$ ...
3
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0
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135
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Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
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Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
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0
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Similarity of non-standard matrices
I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE
$$
u_t+au_x=0.
$$
This is a common approach, because successful methods for this model ...
- 111
2
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0
answers
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A conjectured generalization of Marcus's inequality
Note: I have edited the post below in order to include sharper (conjectured) inequalities, using $|G_1 \cap G_2|$.
Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $...
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vote
2
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146
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Entrywise $\infty$-norm of squared difference of square roots of matrices
For a positive $n \times n$ definite real matrix $M$ we denote by $\sqrt{M}$ the positive square root of $M$. For an $n \times n$ matrix $A$ denote its entrywise infinity norm by
$$\|A\|_{\infty,\...
- 309
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1
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distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
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Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
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0
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Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
1
vote
1
answer
279
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How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
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1
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274
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First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
I am working with the matrix function
$$
f(A) = \frac{1}{\lambda_{\min}(A)},
$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
- 91
2
votes
0
answers
87
views
Nonlinear random matrix equations
Let $C$ be a matrix; $v$ be a column vector;
$P$, $\Delta$ are random matrices;
$x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
6
votes
1
answer
188
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Bound on when sequence of norms of matrix powers starts to decrease
If a matrix $A$ has spectral radius $\rho(A)<1$, it is well-known that $A^n\to0$ as $n\to\infty$, or equivalently $\lVert A^n\rVert\to 0$ for some matrix norm $\lVert\cdot\rVert$; however, it may ...
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1
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1
answer
209
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How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
5
votes
1
answer
255
views
Orthogonal projection onto cones in inner product spaces
Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$,
$$A=\sum_i \lambda_i x_ix_i^*,$$
one can define the positive and negative ...
- 4,503
10
votes
3
answers
582
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
3
votes
0
answers
132
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Bounding the norm of a sum of fourth-order Gaussian vectors
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$, where $\Sigma = \operatorname{diag}(\lambda_1, \cdots, \lambda_k)$ and
$\lambda_1 \ge \cdots \ge \...
- 107
1
vote
1
answer
237
views
Matrix concentration inequality for unbounded (sub-exponential) matrices
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...
- 107
7
votes
1
answer
357
views
Existence of a linear map resulting in the determinant being an elementary symmetric polynomial
Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
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