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 ...

learn more… | top users | synonyms

20
votes
1answer
454 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
0
votes
0answers
19 views

Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem \begin{align} \lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\ &r_i\geq ...
3
votes
1answer
114 views

Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that $\|A\| < x$ and ...
-5
votes
0answers
60 views

Prove determinant of nxn matrix is (a+(n-1)b)(a-b)^(n-1)? [on hold]

Prove det(mat) is (a+(n-1)b)(a-b)^n-1 where matrix is nxn matrix with a's on diagonal and all other elements b, off diagonal? For example, suppose matrix with diagonal composed solely of a's. All ...
0
votes
0answers
72 views

Eigenvalue of (0-1) matrix [on hold]

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
1
vote
0answers
24 views

Trace of Inverse matrix from Cholesky

This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case. I have the matrix $\Sigma=LL^T$. Is there ...
-1
votes
0answers
44 views

vector-matrix notation and expectation of matrix and Hermitian product [closed]

Let $\textbf{h} \in \mathbb{C}^{N\times 1}$, $\textbf{a} \in \mathbb{C}^{N\times 1}$, $\textbf{b} \in \mathbb{C}^{N\times 1}$ and $\textbf{c} \in \mathbb{C}^{N\times 1}$. The variable $h_i$ is defined ...
2
votes
1answer
110 views

About partial uniqueness of SVD

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let ...
3
votes
0answers
126 views

Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
0
votes
0answers
42 views

Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{ii}|-\sum_{j \neq i}|a_{ij}|}$, where the ...
3
votes
1answer
58 views

submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
5
votes
1answer
86 views

An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...
7
votes
0answers
158 views

Solving $P=AB,Q=BA$, in the unknowns $A,B$

Let $p\geq q$ $P\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. We seek $A\in M_{p,q},B\in M_{q,p}$ s.t. $P=AB,Q=BA$. The NS conditions for the existence of $(A,B)$ are given in On the matrices AB and BA. ...
6
votes
2answers
463 views

Is this Hankel matrix in trace class

Let A be the infinite Hankel matrix with the coefficient $$A_{kj}=e^{(-t(k+j)^2)}-e^{(-t(k+j+2)^2)},$$ with $t$ a nonnegative real number. Is $A$ in trace class with a norm bounded by an absolute ...
0
votes
1answer
69 views

Understanding the derivation of a ML-estimator (statistics)

I'm trying to understand the derivation of a ML-estimator and more specifically the rewriting of the covariance matrix $\Sigma$. In this rewriting, a lemma is used to show that: $$ \tag{1} ...
2
votes
1answer
72 views

Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$, the sequence $(M^n)_n$ converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...
0
votes
1answer
50 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
194 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} ...
6
votes
1answer
200 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
78 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
80 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
127 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
236 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 ...
1
vote
0answers
24 views

Conditional Distribution of Inverse Wishart

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
88 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
121 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
137 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
43 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
166 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
84 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
98 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
209 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 ...
2
votes
0answers
61 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
51 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
96 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
69 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
75 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
112 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
129 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
176 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
74 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
58 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
108 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
1answer
127 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
324 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
63 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
240 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
114 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
160 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
1answer
142 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 ...