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
33 views

canonical decomposition of a linear map

I have a linear map $A : {\mathbb F}^k \to {\mathbb F}^{k+m}$ where ${\mathbb F}$ is some field; (you can impose restrictions on ${\mathbb F}$ if it helps). I'd like to decompose $A$ as a product of ...
-2
votes
0answers
43 views

concentric spheres with common radius [on hold]

I am trying to think of a way to solve this problem but I am not sure if it is doable or not. Here goes: Assume we have n spheres that share a common radius (x0,y0,z0). For each sphere we have one ...
-2
votes
0answers
22 views

Find the number of connected components in pseudospectra [on hold]

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 ...
-2
votes
0answers
10 views

Best algorithm to compute the first eigenvector of symmetric matrix [migrated]

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
-5
votes
0answers
13 views

applications of systems of linear equations [on hold]

A person plans to invest a total of ​$260,000 in a money market​ account, a bond​ fund, an international stock​ fund, and a domestic stock fund. She wants 60​% of her investment to be conservative​ ...
1
vote
0answers
56 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 ...
-5
votes
0answers
38 views

Maple: In Matrices A x B = C, how do I find matrix A given B and C [on hold]

I have matrix A, B, and C which are all 8x8 matrices in Maple. in the equation A x B = C, when B and C are known, how do I find matrix A? I know how to do it by hand, but I don't know the maple ...
4
votes
1answer
73 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
53 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?
1
vote
0answers
55 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 ...
-3
votes
1answer
55 views

A question on matrix polynomial [closed]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
0
votes
0answers
26 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
42 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 ...
1
vote
0answers
35 views

How to calculate the derivative of logarithm of a matrix? [migrated]

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(1-M)^k}{k}$$ The derivative of ...
4
votes
2answers
127 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 $A+cI$, ...
0
votes
0answers
13 views

Compactness and Convexity of space of correlation matrices [closed]

A $n\times n$ real symmetric matrix is a correlation matrix, if it is positive-semidefinite and all its diagonal entries equal 1. According to most references it is easy to see that the space of ...
0
votes
0answers
12 views

Efficient Row Sum of Factorized Matrix [closed]

I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix. The matrix was factored using SVD: A -> U, S, V -> U, ...
2
votes
1answer
90 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 ...
9
votes
0answers
228 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
25 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 ...
0
votes
0answers
17 views

Norm of a linear operator in a tight frame

My question certainly has a simple answer, but I am not sure about how to formalize my thoughts, to put it simply, I am looking for the norm of a linear operator that is a composition of 2 linear ...
2
votes
1answer
150 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
145 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
169 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
75 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
100 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
36 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 ...
1
vote
1answer
91 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
372 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
77 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
25 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
33 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
64 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
170 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: ...
0
votes
0answers
9 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
78 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
113 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
104 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
40 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 ...
7
votes
2answers
285 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 \\ ...
2
votes
1answer
157 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
68 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
64 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
74 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
103 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
336 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
55 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, ...
0
votes
0answers
234 views

When does this matrix have full rank?

Suppose $\mathbf{B}\in\left[0,1\right]^{T\times M}$ is a binary matrix, $\mathbf{B}_{i}$ is a column of $\mathbf{B}$, and $\mathbf{X}\in\mathbb{R}^{N\times T}$ is a matrix where the columns are ...
0
votes
2answers
66 views

Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that ...
0
votes
1answer
36 views

Making a real matrix positive definite by replacing nonzero and nondiagonal entries with arbitrary nonzero reals

Let $A$ be real square matrix. Let $\mathcal{F}(A)$ be the set of real matrices $A'$ of the same size such that $A'_{ii}=A_{ii}$ for all $i$, and for all $i,j$, $A_{ij}=0\Rightarrow A'_{ij}=0 \land ...