Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level ...

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0
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0answers
14 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
-4
votes
0answers
27 views

Find solution of signle element $y_i$ in vector $y$ subject to $Ay=c$ [on hold]

I have a interesting question about linear algebra problem. Assume that I have a matrix $A^{m \times n}$ and vector $c^{n \times 1}$ are known and I want to find the solution of vector y subject to ...
-4
votes
0answers
23 views

matrix summation and uniform convergence of matrix sequence [on hold]

I start studying summation theory and as we know a matrix is called a Schur matrix if i)lim (a)nk= x when n goes to infinity and ii)the sum of (a)nk that means the convergence is uniform in n. can ...
2
votes
1answer
69 views

Unitary factor in polar decompositions

Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant ...
1
vote
0answers
119 views

Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x ...
1
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1answer
47 views

Efficient computation of null space of large symbolic matrices?

Are there any computer algebra system/libraries that can compute the null space of a large symbolic matrix in parallel? This problem arises when finding invariant polynomials of a continuous linear ...
1
vote
0answers
80 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
7
votes
2answers
245 views

Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough. What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
-3
votes
3answers
276 views

Determinant of matrix from set {-1, 1} [on hold]

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$? I have ...
1
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0answers
25 views

Modified Orthonormal Procrustes Problem

In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and ...
3
votes
1answer
50 views

Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...
1
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0answers
33 views

Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...
3
votes
0answers
29 views

Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup

I tried asking this on math exchange, but no luck, so thought I'd try here. Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
0
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0answers
18 views

Composition of Lossless Systems from Delay and Mixing regarding junction admittance

Given $m_1, \dots m_N \in \mathbb{N}$ and matrix $\mathbf{A} \in \mathbb{C}^{N\times N}$. Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{-m_1}, \dots, z^{-m_N})$ with $z\in\mathbb{C}$. ...
5
votes
1answer
141 views

A coalgebra structure on compact operators

Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus ...
2
votes
0answers
75 views

Copositivity under tensor products

Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries? ...
3
votes
0answers
108 views

Commutative decomposition for full-rank $A$ and low-rank $B$ matrices that do not commute

1. Motivation Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following ...
11
votes
0answers
223 views

Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - ...
5
votes
2answers
253 views

Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name. That is, a matrix (here ...
38
votes
1answer
731 views

Determinant of a determinant

Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m ...
5
votes
0answers
156 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
5
votes
1answer
181 views

Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows $$ B_6=\begin{bmatrix} & & & & & 6\\ & & & ...
5
votes
1answer
479 views

Proving that the kernel of this matrix is of dimension 2

(Edit : see at the bottom of the question for an additional surprising possible hint.) Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when ...
1
vote
1answer
60 views

Expected value of the inverse of a random, truncated Haar matrix

Let $Q$ be a (say 4x4) unitary matrix, distributed according to the Haar distribution. Denote the upper left 2x2 submatrix of $Q$ as $Q_{1:2,1:2}$. I am interested in the following expectation: $E(I ...
-1
votes
1answer
43 views

How to construct a semi-positive definite matrix in this form: (L=D-A')

As known, the graph Laplacian $L = D - A$ is semi-positive definite. What if there is a matrix $A'$ where $$ A'_{ij} = \begin{cases} A_{ij}, \quad if A_{ij} >0 \\ -\varepsilon, \quad if A_{ij} = ...
3
votes
1answer
132 views

the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form \begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} ...
0
votes
0answers
37 views

Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
0
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0answers
20 views

Is there effective algorithm for finding “minimal discovery time” for large graphs?

Consider a large, probably sparse graph with Markovian random walkers on it. Define discovery time as time to first reach a vertex by random walk from uniform start. Are there effective ways to find ...
2
votes
1answer
105 views

Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix $V = \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right)$ has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = ...
2
votes
0answers
59 views

Is there an efficient way to compute the “complete subset regression”?

Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...
1
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0answers
21 views

How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference

An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...
4
votes
2answers
114 views

The eigenvectors and eigenvalues of matrix geometric mean

This is a follow up question on from How to solve a non-homogeneous quadratic matrix equation?. Given the matrix $G = A(A^{-1}M)^{1/2}=A^{1/2}(A^{-1/2}MA^{-1/2})^{1/2}A^{1/2}$, where $A=-H^{-1}$, ...
4
votes
1answer
106 views

Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices

Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...
1
vote
1answer
44 views

Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...
0
votes
0answers
20 views

Computing covariance matrix from patchy data

Take a linear Gaussian model given by the following normal distributions: $\mathcal{N}(z_n | z_{n-1}, A), \ \ \ \ \ \ \ \ \mathcal{N}(x_n | z_{n}, B) $ Let variable $z$ be a hidden variable and $x$ ...
1
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0answers
72 views

Does this permanent have a closed form?

What is the closed form of this permanent? (similar to the Cauchy determinant) \begin{aligned} f(z_1,z_2,\cdots,z_N,w_1,w_2,\cdots,w_N)=\left[ \small{\begin{matrix} \frac{1}{(z_1-w_1)^2} && ...
4
votes
1answer
215 views

Generalized Cauchy-Binet sum over a fixed subset of indices

I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
3
votes
1answer
70 views

Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries $$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$ where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...
1
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0answers
75 views

MInors related problem [closed]

A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
2
votes
0answers
64 views

If matrices describe simplices, what do matrix operations describe?

Suppose we are given a $d \times d$ matrix $M$ with rows $m_1, \dots, m_d$. This matrix describes a simplex, namely the convex closure of the origin with the vectors $m_1, \dots, m_d$. Now, scaling ...
5
votes
3answers
202 views

Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
11
votes
1answer
201 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
2
votes
0answers
157 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
1
vote
0answers
42 views

a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
2
votes
1answer
100 views

Handelman's positivstellensatz for symmetric matrix-valued polynomials

For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$. Several ...
-1
votes
1answer
84 views

Action of rotation group on Matrices [closed]

Is the following assertion true? Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in ...
9
votes
1answer
257 views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
0
votes
1answer
86 views

Eigenvalues of product of diagonal positive matrix and symmetric matrix [closed]

Assume that we have two real symmetric matrices A and B, where A is a positive diagonal matrix, and B is a symmetric matrix with one eigenvalue λ = 0. Assume that H= AB; is it possible to proof that ...
10
votes
3answers
627 views

A class of matrix determinants between Wronskians and Vandermondes

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions ...
4
votes
1answer
187 views

Another kind of the positivity of matrices

For given $n$, the following $n\times n$ complex matrix $M=M^{\dagger}$ is called positive, if $x^{\dagger}M x\geq 0$ holds for all complex vector $x=(z,z^2,\cdots,z^n)^T$ with arbitrary complex ...