Questions tagged [matrices]

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 tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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3
votes
1answer
66 views

Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
0
votes
1answer
71 views

Isomorphism between two groups [closed]

Let $\mathbf{F}_q$ be finite field of order $q$, where $q$ is an odd-prime (or power of an odd prime). Is there an isomorphism between the following subgroups of unipotent $3\times 3$ matrices over $\...
4
votes
0answers
51 views

Characterization of “PSD-Squared” Matrices

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
0
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0answers
23 views

Is the SVD optimal (by Eckart Young theorem) because maximal data variance is captured from both the column and row space?

I am asking this question seeking validation of an intuitive understanding of the veracity of the Eckart-Young theorem which struck me in my study of the SVD and Principle Component Analysis. The ...
-1
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0answers
63 views

What is the normal subgroup generated by nonsingular, positive definite matrices? [closed]

Within $GL(n,\mathbb C)$ let $H$ denote the normal subgroup generated by products of positive-definite, Hermitian matrices. What is this subgroup? And what is the quotient $GL(n,\mathbb C)/H$? Do ...
1
vote
1answer
102 views

Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
6
votes
0answers
79 views

An integral Jacobson-Morozov theorem?

$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \...
1
vote
1answer
140 views

Do there exist graphs whose adjacency matrix is positive semi-definite? [closed]

If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
-1
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0answers
37 views

“euler form” of a unitary matrix? [closed]

A little background: I'm a student of physics (forgive me for a difference in language or the lack of familiarity with this topic), trying to solve an equation of the form: \begin{equation} \mathcal{...
0
votes
0answers
37 views

trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form? Such an expression would naturally generalise some invariant random matrix ensembles. ...
1
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0answers
88 views

Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$: $$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...
27
votes
0answers
441 views
+100

Sum over 0-1 matrices

I stumbled across the following formula when working on a research problem in theoretical computer science. I am looking for a simple proof of it, or any idea which might prove useful. I checked its ...
12
votes
0answers
261 views

Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field

Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
6
votes
2answers
222 views

Characteristic polynomial of checker matrix

For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...
4
votes
0answers
214 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
4
votes
1answer
154 views

Neighborhood of an orthogonal matrix

Let $A\in O(n)$ be an orthogonal matrix and let $\vec{a}_1,\dots,\vec{a}_n$ be its rows. For a vector $\vec{v}=[v_1,\dots,v_n]$, let $\max(\vec{v})=\max\{|v_1|,\dots,|v_n|\}$. Prove or disprove that ...
0
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0answers
32 views

pseudo-inverse of short fat matrix

Consider the matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$. Is its pseudo-inverse $A^\dagger = (A^\top A)^{-1} A^\top$ computable ? I'd expect not, because $A^\top A$ is not of full rank ( $...
4
votes
1answer
246 views

Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor

I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...
-2
votes
0answers
57 views

Can we use modified Cholesky's decomposition for solving linear equations?

So I was learning about modified Cholesky's decomposition, specifically Gill, Murray, and Wright algorithm. And although I was able to decompose a negative definite matrix into $LDL^T$, when I used ...
3
votes
1answer
146 views

An orthogonal matrix that satisfies a property must be a permutation matrix

Let $A$ be an $n\times n$ orthogonal matrix such that $\sum_{k=1}^na_{ik}^3a_{jk}=\sum_{k=1}^na_{jk}^3a_{ik}$ for every $1\le i,j\le n$. Original question which is solved by a counterexample given (...
2
votes
1answer
121 views

Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S

Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
1
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0answers
28 views

Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements

Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric) $$ \begin{...
4
votes
0answers
109 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
8
votes
0answers
117 views

Membership problem in general linear group

This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am. Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...
1
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0answers
41 views

Diagonally similar to submatrix of orthogonal matrix

Given $A \in \mathbb{R}^{n \times n}$, with $0 < |\det(A)| < 1$. Does a diagonal matrix $D$ exist such that $$ B = D^{-1} A D $$ is the principal submatrix of an $(n+1)\times(n+1)$ orthogonal ...
1
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0answers
35 views

Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem: I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$. The first is a regular Toeplitz matrix $A$...
2
votes
1answer
57 views

Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices

For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices: $$\Pi(A)=\mathrm{argmin}_{M\...
3
votes
2answers
157 views

Maximize $l_1$ norm with unitary matrix

Given an invertible matrix $A \in \mathbb{C}^{n\times n}$. How to find $$ U^* = \max_{\text{$U$ with $U^H U = I$}} \lVert U A\rVert_1, $$ where $\lVert\cdot\rVert_1$ is the entrywise 1-norm, i.e., $\...
3
votes
0answers
31 views

A non-singularity property for sets of real matrices

Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
0
votes
1answer
149 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
0
votes
0answers
40 views

“Probability” for a partitioned matrix to be singular

Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix $$ M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
0
votes
0answers
40 views

Find occurrences of certain matrix inside a matrix

This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching. We know for ...
5
votes
1answer
259 views

How to compute a more general version of Vandermonde / Cauchy double alternant determinant

Consider some variables $\{X_i\}_{1\le i \le n}$, $\{Y_i\}_{1\le i \le n}$, and $\{W_i\}_{1\le i \le n}$. Does anyone know how to compute the following determinant? $$ \det ~ \left(\frac{W_j^{i-1}}{...
1
vote
1answer
74 views

Low-complexity method for sub-matrix inversion

Assume that $\mathbf{A}$ be an $N\times N$ matrix. We know that the complexity of the computation of matrix inversion is $\mathcal{O}(N^3)$. Let define $\mathbf{D}=\mathbf{A}^{-1}$. Now, assume that $\...
16
votes
2answers
978 views

How to prove the determinant of a Hilbert-like matrix with parameter is non-zero

Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero? $$ \begin{pmatrix} \frac{1}{\...
9
votes
3answers
376 views

Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices

I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a ...
4
votes
1answer
105 views

Counting adjacency matrices

Here is a question that has come up in the context of a problem that involves counting partially ordered sets. For an adjacency matrix $A$, let $p$ be the sum of elements in the strict upper ...
3
votes
0answers
106 views

Diffeomorphisms of a “matrix type”

Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form $$ f(x) = \exp\left( \sum_{i=1}^n f_i(x) A_i \...
1
vote
0answers
28 views

Probability of marking at least one row in given matrix

Let there be a matrix $\alpha=(a_{i,j})_{i\in [m], j\in [n]}$, where $a_{i,j}\in\{0,1\}$ And every row has exactly $r\le n$ ones. We independently with probability $p$ choose some columns from this ...
4
votes
1answer
107 views

Relation of row sums to largest eigenvalue

I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
0
votes
0answers
27 views

Generalized eigenvectors product

Let's consider a real square matrix $A$ with eigenvalues $\lambda_n$ and eigenvectors $\mathbb x_n$, i.e. $A \mathbb x_n = \lambda_n \mathbb x_n$. Suppose there are some generalized eigenvectors $\...
0
votes
2answers
81 views

Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli

Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$. (Note we must have $1-p_1-p_2+p\ge 0$ for the ...
12
votes
2answers
652 views

Abelianization of general linear group of a polynomial ring

For $K$ a field, is it known what the abelianization of $GL_2(K[X])$ is?
0
votes
1answer
60 views

Maximize function on rotation matrices [closed]

Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively: \begin{equation} Q= \...
5
votes
1answer
84 views

When is the log-permanent concave?

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
1
vote
1answer
152 views

A linear algebra question regarding the eigenvalues of the product of a diagonal matrix and a projection matrix

I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few ...
0
votes
1answer
32 views

The product of non-primitive matrices with zero positions in common

Def.[1]: A non-negative $n \times n$ matrix $A$ is called a non-primitive if there is no an integer $k$ such that all entries of $A^k$ are positive.[1] Def.[2]: Let ${\bf A}=(a_{i,j})$ and ${\...
1
vote
0answers
34 views

Minimum rank of a product of two block diagonal matrices with an arbitrary matrix

Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
6
votes
0answers
187 views

Computing powers of a special matrix fast

I've got an $n\times n$ matrix that is upper-triangular (all zeros below diagonal), diagonal entries are positive, and entries that are not on the diagonal are either $0$ or $1$. An example matrix ...
1
vote
0answers
78 views

Abelian subgroup of maximal order

Let $\mathbf{F}_q$ be a finite field of order $q$ ($q$, an odd prime, or a power of the same). I know that for the matrix algebra $M(n,\mathbf{F}_q)$ (or $gl(n,q)$), the maximal dimension for a ...

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