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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-1
votes
1answer
66 views

Strassen's algorithm for higher order matrices [closed]

So I am clear with the basic Strassen's algorithm. For a regular 2*2 matrix, it will perform 7 multiplications instead of the conventional 8 for regular matrix multiplication algorithms. The seven ...
1
vote
2answers
204 views

What is the name for the type of matrices?

Let $ K $ be a field. We can recursively define matrices as $ M_{a} = (a)$ for any $ a\in K $ and $$ M_{a_1, \cdots, a_{2^i}} = \begin{pmatrix} M_{a_1, \cdots, a_{2^{i-1}}} & M_{a_{2^{i-1} ...
8
votes
1answer
285 views

An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have $$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
2
votes
1answer
83 views

Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this? ...
2
votes
0answers
95 views

Stationary Distribution for Markov-like system?

Let \begin{equation} A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix}, \end{equation} \begin{equation} B= ...
4
votes
2answers
234 views

Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...
1
vote
1answer
330 views

Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$

The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ...
0
votes
1answer
98 views

Conditional Distribution of Inverse Wishart [closed]

Suppose $\begin{bmatrix} K_{11} K_{12}\\K_{12}^T K_{22} \end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix} \Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22} \end{bmatrix}\right)$. What is the ...
3
votes
1answer
107 views

Trace of multiplied positive definite matrices

I have to compute $Tr(K^{-1}\Sigma)$ where both $K$ and $\Sigma$ are symmetric positive definite matrices. Question is considering that I have computed the Cholesky, $L_1$ of $K$ previously, is there ...
-2
votes
1answer
149 views

Solving a difficult equation for a variable?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
2
votes
0answers
250 views

Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where, $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ $A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$. $B \in ...
0
votes
0answers
59 views

Eigenvalues of a “Half-Kronecker ” Product

The Problem: Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other 2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given). Now we can construct a new matrix ...
2
votes
1answer
298 views

Matrix equation XAX=B where the solution must be diagonal [closed]

$$X_{solution}=\arg\min_X \|XAX-B|_F \quad\mathrm{subject\ to}$$ X is square and diagonal A is square and positive semi-definite B is square and positive semi-definite Any pointers or relevant ...
-1
votes
1answer
115 views

Determinant of a sum of two Hankel matrices [closed]

First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? No. Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if ...
3
votes
1answer
109 views

The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$. This situation has many different names: $G$ is called the cone or the ...
10
votes
1answer
467 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
14
votes
1answer
489 views

A matrix completion problem

In their paper, "Corners of Normal Matrices," R. Bhatia and M.D. Choi ask the following question: Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such that the block ...
0
votes
0answers
60 views

Property of quasipositive matrices

I saw this theorem stated in a paper without proof and I have difficulty proving it. If $A$ is an $n\times n$ matrix with non-negative off-diagonal entries, let $s(A)$ be the real eigenvalue such ...
1
vote
2answers
106 views

Mixing Numerical Range and inner product

Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as \begin{align} ...
-3
votes
1answer
116 views

Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ? To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
4
votes
0answers
89 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
3
votes
1answer
125 views

Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$. Define the ...
6
votes
2answers
146 views

How to write this result (successive Schur complements compose nicely)

There is an easy theorem in linear algebra that "successive Schur complementations compose nicely": for instance, let's say I have a matrix $$ M_0= \begin{bmatrix} A & B & C \\ D & E & ...
3
votes
1answer
194 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
1
vote
0answers
333 views

Inverse Transpose of Jacobian Matrix

Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by \begin{equation} f(x)\approx ...
1
vote
0answers
70 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
2
votes
1answer
154 views

derivative of sum of singular values

can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix? I am trying to minimize $$\min_A \parallel A \parallel_*+ \cdots $$ where $\parallel A ...
2
votes
2answers
207 views

Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
5
votes
1answer
396 views

Lights out game

I would like to ask about the game Lights Out for a square nxn. In http://mathworld.wolfram.com/LightsOutPuzzle.html there is a list of the number of solutions to the game, and the number of solutions ...
2
votes
2answers
73 views

Markov-type functions

I'd like to have some informations about Markov-type functions (or Cauchy-type): \[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\] $\gamma$ is a positive measure with compact support ...
6
votes
1answer
274 views

Injectivity of matrix “fingerprint”

Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries. For any matrix $A$, define $$ ...
1
vote
0answers
257 views

Closed-form solution to a system of linear equations

Consider the following $n \times n$ matrix with a particularly nice structure: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 &1\\ 0 & 0& \dots&0 & ...
6
votes
1answer
177 views

Almost Hadamard matrices

As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$ and pairwise orthogonal rows or columns. Such matrices exist conjecturally in every dimension divisible by $4$. Call ...
3
votes
2answers
198 views

A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$

Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of ...
3
votes
1answer
245 views

Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if $$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...
1
vote
0answers
180 views

Incoherence of the row/column span

Due to V.Chandrasekaran., et al‎ (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that: $$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$ where the lower bound is achieved (for ...
1
vote
1answer
230 views

$SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$. I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
14
votes
4answers
723 views

Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 ...
-1
votes
1answer
163 views

Inequality between two matrices

Given a full rank $n \times m$ matrix $K$ with $m<n$ and an invertible symmetric matrix $J$. Let $A$ be a symmetric positive semi-definite $n \times n$ matrix such that \begin{equation} (K^T ...
4
votes
1answer
309 views

Eigenvalues of generalized Vandermonde matrices

Given a strictly increasing sequence $0<x_1<x_2<\dots<x_n$ of $n$ strictly positive real numbers and a second strictly increasing sequence $e_1<\dots e_n$ of $n$ real numbers, the ...
9
votes
1answer
319 views

Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 ...
0
votes
1answer
343 views

Whether R has a Cholesky decomposition when R is a positive semi-definite?

It is no doubt that R has a Cholesky decomposition when R is a positive definite matrix.I want to ask Whether R has a Cholesky decomposition when R is a positive semi-definite?I would appreciate it if ...
1
vote
1answer
261 views

Random matrix determinant problem

Suppose we have a a set of random matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are random complex valued ...
0
votes
1answer
152 views

A determinant problem with symmetric PSD matrices

Suppose we have a a set of matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are constant complex valued finite ...
9
votes
4answers
438 views

When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter. Now consider the family of representations ...
0
votes
1answer
109 views

Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} ...
6
votes
1answer
158 views

A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation. A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...
1
vote
0answers
260 views

how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying ...
3
votes
3answers
233 views

Applications of rank factorization or full rank decomposition [closed]

I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r ...
0
votes
1answer
113 views

Kernel of a projection

Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define ...