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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19
votes
0answers
578 views

Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
0
votes
0answers
31 views

Distribution of zeros and ones over matrix

I have the following problem: Given a matrix with n rows and m columns. Some elements of the matrix are unavailable. For each column, you have a set containing a number of zeros and ones which must ...
2
votes
0answers
135 views

On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration \begin{equation} X_{k+1}=\frac{1}{N}\sum_{i=1}^...
7
votes
1answer
82 views

Add a multiple of $I$ to a matrix to minimize its operator norm

Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$? Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity. The corresponding problem for the ...
2
votes
0answers
169 views

Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\...
1
vote
0answers
83 views

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
0
votes
1answer
114 views

Derivative of matrix logarithm w.r.t. real parameter

Let $A(t)$ be an invertible square matrix that depends (differentiably) on a real parameter $t$. It is well known that for example $$ \frac{d}{dt} A(t)^{-1}=-A(t)^{-1}\ \dot{A}(t)\ A(t)^{-1} $$ and $$ ...
7
votes
3answers
200 views

Is there a standard name for the following type of linear operator?

Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
2
votes
1answer
255 views

A presentation for $GL(2,\mathbb{Z}/p^n \mathbb{Z})$

In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$: $\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = ...
1
vote
0answers
91 views

Roots of matrices in $G_2(Z)$

Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...
7
votes
1answer
758 views

The saturation of Murray von Neumann relation

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version: Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows: We say $a M b$ iff $...
9
votes
2answers
255 views

Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...
4
votes
0answers
62 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
2
votes
0answers
137 views

Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$? Is it enough to consider the injection via Hochschild: $Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$? Edit: The ...
1
vote
0answers
100 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
6
votes
1answer
109 views

Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions? To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
2
votes
0answers
75 views

Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
2
votes
0answers
99 views

Computing abelianizations of some explict finite subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$

I have been attempting to find some abelianizations of some subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$. I have been using brute force for the most part but I get very messy results. Here is an ...
0
votes
1answer
60 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
2
votes
1answer
80 views

Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix $$\tilde{Q}\,=\,(1-\alpha)Q+\...
4
votes
2answers
197 views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
2
votes
1answer
105 views

Similarity via symmetric matrix

Let $K$ be a field extension of $F$. If two $n\times n$ matrices $A,B \in M_n(F)$ are similar via a matrix $P \in GL_n(K)$ (that is, $A=PBP^{-1}$), then there exists a matrix $Q\in GL_n(F)$ such that $...
10
votes
2answers
427 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...
0
votes
0answers
45 views

Nuclear norm maximization

I am trying to solve a nuclear norm maximization problem: $$\arg \max_{Q \in O(n)} \|WQV^T\|_*$$ where $Q$ is an $n \times n$ orthogonal matrix and $W$ and $V$ are real $d \times n$ matrices. I've ...
2
votes
1answer
166 views

Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...
3
votes
1answer
155 views

Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$ Let $M_n$ be the space of $n \times n$ real matrices. Question: For which $n$, is there an inner product on $M_n$ which satisfies: $$(*) \, \, \langle Q^TXQ,Q^TYQ \...
3
votes
2answers
198 views

functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
4
votes
1answer
83 views

Kolmogorov complexity for matrices

In applications one often encounters very large matrices that barely fit in computer memory, if at all. Naturally one wishes to represent those matrices as compactly as possible. Sometimes one even ...
4
votes
0answers
115 views

Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$

Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
0
votes
1answer
41 views

Cross section point of two conics curves

We have $A_i , B_i , C_i , D_i , E_i ,F_i, \ (i=1, 2) $. We want to find $ (u,v) \in \mathbb{R}^2$ satisfying \begin{equation} A_1 u^2 + B_1 uv + C_1 v^2 + D_1 u + E_1 v +F_1 =0 \\ A_2 u^2 + B_2 ...
2
votes
1answer
151 views

the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
11
votes
1answer
472 views

An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
0
votes
0answers
84 views

Primitivity of $AA^\top$

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...
0
votes
0answers
31 views

A class of unimodular parametrization

Is there a parametrization of set of matrices $\mathcal M\subseteq\Bbb Z[x_1,\dots,x_{m}]^{n\times n}$ such that $\forall f:\{-1,+1\}^{m}\rightarrow\{-1,+1\}$ $\exists M\in\mathcal M$ such that $\...
0
votes
0answers
40 views

If $A$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$

If $A \in M_n$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$, in which both $A_1$ and $A_2$ are doubly stochastic?
1
vote
0answers
75 views

Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?

Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix $M = \frac{1}{2}(A + {A^T})$. Why does $\rho (A) \le {\lambda _{\max }}(M)$?
-1
votes
1answer
177 views

FInd smallest value $r$ such that a $n\times r$ matrix exists [closed]

The input of my problem is an integer $n\geq 3$. The output is an integer $r\geq 1$ which must be as small as possible such that there is a $(n\times r)$ matrix verifying the following constraints: ...
1
vote
0answers
18 views

How can I filter the effects of a variable from a correlation matrix?

I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...
1
vote
1answer
79 views

Multiplicatively closed subsets of $\mathbb{C}^{n \times n}$

While dealing with another problem I saw that I need to classify subspaces $V$ of $\mathbb{C}^{n \times n}$ that are multiplicatively closed. So to get an idea of the nature of the subspaces I ...
0
votes
1answer
120 views

Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
0
votes
1answer
106 views

Number of turning points on a nondecreasing $n^2 \times n^2$ matrix

Given a integer value $n$, we generate a $n^2 \times n^2$ integer matrix $M$ in the following way. Each ceil has value range $[1~n]$ In each row, the value is nondecreasing. E.g. $M[i, j] \leq M[i, ...
1
vote
0answers
63 views

Decomposing a matrix into the tensor product of a permutation and orthogonal matrix

Suppose I have a square matrix $A \in \mathbb{R}^{mn \times mn}$. I want to find $$\arg \min_{P, Q} \|A - P \otimes Q\|_F$$ where $P$ is an $m \times m$ permutation matrix and $Q$ is an $n \times n$...
7
votes
2answers
298 views

Covering the zeros of 0/1 matrix with submatrices

The matrices I am dealing with are $n\times n$ of the following type (with $n=7$): $M_7=\begin{pmatrix}1&0&0&0&0&0&1 \\ 1&1&0&0&0&0&0 \\ 0&1&1&...
3
votes
1answer
182 views

On a determinantal equality

In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here). Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is ...
1
vote
0answers
76 views

A question on Perron–Frobenius theorem [closed]

Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$). Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries. Why is there a permutation ...
4
votes
1answer
74 views

Robust generalization of matrix rank

I am looking for robust generalizations of matrix rank. Think of the the following problem: A big matrix of low rank is perturbed by random noise, such that it becomes a full-rank matrix. Is there a ...
2
votes
1answer
100 views

Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic, linear operators, which says the following: Defintion: Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...
4
votes
1answer
147 views

What is the best algorithm for even rank magic square?

Magic square is a $n*n$ matrix with numbers of $1,2,...,n^2$ and has the property that sum of any row and any column and sum of main diameter and adjunct diameter is identical. There exists a very ...
5
votes
2answers
385 views

Does the antidiagonal in this square matrix always contain a prime?

Does the antidiagonal in the square matrix with entries $1,2,\ldots,n^2$ row by row in that order always contain a prime? For example: For n=2: $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ ...
3
votes
1answer
99 views

Similarity transformation of transition matrix of reversible Markov chain (reference request)

If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by: $$R_{ij} = \sqrt{\frac{\pi_i}{\pi_j}}P_{ij}~.$$ By reversibility, ...