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.
3,315 questions
1 vote
1 answer
130 views
How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
- 75
45 votes
7 answers
2k views
If $\det(M)=ab$ is it true that $M=AB$ with $\det(A)=a, \det(B)=b$?
For which (commutative) rings $R$ and dimensions $n$ is the following claim true (or false)? Claim: For all $a,b$ any $n\times n$ matrix $M$ with coefficients in $R$ and $\det(M)=ab$, can be factored ...
- 5,815
2 votes
1 answer
182 views
A Loewner ordering problem
Let $A$ be a positive definite diagonal matrix, $B$ be a real matrix, and $C$ be a complex matrix. All are square matrices of dimension $n$. I am wondering if it's true that $$\Big\|A+B^\top C^* C B\...
- 71
3 votes
1 answer
322 views
For which $k$ does a generic choice of $k$ $n \times n$ matrices span a subspace of $\mathrm{GL}(n)$?
If $\rho(n)$ are the Radon-Hurwitz numbers, then for $k \leq \rho(n)$ it is possible to find $k$ $n \times n$ real-valued matrices $A_1, \dots, A_k$ so that for any $(a_1,\dots,a_k) \neq 0$, the ...
- 507
0 votes
2 answers
183 views
Elementwise unreachable matrix
Let $A \in \mathbb{F}^{n \times n}$ be a square matrix, and let $(i,j)$ denote the entry in the $i$-th row and $j$-th column of $A$. We say that the position $(i,j)$ is unreachable if for all positive ...
- 11
-6 votes
1 answer
115 views
Determining if binary matrix with specific form has full rank [closed]
I have the following 15x15 binary matrix with a specific form: $$\begin{bmatrix} 1&1&0&0&0&0&1&1&1&1&1&1&0&0&0 \\ 1&0&1&0&0&...
- 1
13 votes
1 answer
370 views
Joint spectrum of two matrices and simultaneous upper triangulisation
I consider the following conjecture: Let $A,B$ be $n\times n$ matrices over $\mathbb{C}$ (or any algebraically closed field of characteristic zero). The following are equivalent: $\det(I+xA+yB)\in\...
- 882
0 votes
0 answers
92 views
Variant of Cordes Inequality
The classical Cordes inequality states the following: suppose that $\|\cdot\|$ is the usual matrix norm, $0 < \alpha \leq 1$, and $A, B$ are $n \times n$ positive semidefinite Hermitian matrices. ...
-1 votes
2 answers
117 views
Constructing an orthonormal set with given projections in a direct sum decomposition
Let $V$ be an $n$-dimensional real inner product space. Suppose we have $k\leq n/2$ orthonormal vectors $u_1, u_2, \dots, u_k \in V$. Assume that there exist pairwise orthogonal subspaces $A,B,C \...
- 11
0 votes
0 answers
141 views
On tensor product and rank
I am getting confused by the tensor product. I would appreciate some basic insight. I consider $M_2\otimes M_3$. (Here $M_n$ denotes the complex $n\times n$ matrices.) The dimension of this space is 4*...
- 109
0 votes
0 answers
132 views
Matrix factorizations under quotient ring
Let $(R, \mathfrak{m})$ be a regular complete local ring, let $f \in \mathfrak{m}^2$, and let $b \in \mathfrak{m}$ such that $b$ is a non zero divisor of both $R$ and $R/\langle f \rangle$. Denote $S=...
2 votes
0 answers
130 views
Lipschitz property of Frobenius norm of "standard deviation matrix"
Question: Is the Frobenius norm of (some form of) standard deviation matrix Lipschitz with respect to the Wasserstein distance? To be more precise: Suppose $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ are two-...
- 21
2 votes
1 answer
193 views
Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that $$ \underset{P^2 = P,\; \text{rank}(P) = p}{...
- 23
2 votes
0 answers
149 views
I am looking for "something like" an entry-wise matrix 1/2-norm. Has such a thing been studied? Where should I look?
Suppose an $n\times n$ square matrix $A$ with real or complex entries $A_{ij}$. Now define a quantity $Z(A)$ associated with the matrix by $$ Z(A)=\sum_i \sum_j |A_{ij}|^{1/2}. $$ What is this ...
0 votes
0 answers
83 views
Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem stems from a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the similarity transforming matrix from ${\bf J}$ to ${\bf ...
- 75
1 vote
1 answer
398 views
Sum of ranks of blocks
Let $M = \pmatrix{A & B\\ C& D}$, where $A$ is an all-one matrix. From Section 3 of Nisan & Wigderson$\color{magenta}{^\star}$, $$\operatorname{rk} (B) + \operatorname{rk} (C) \le \...
- 521
7 votes
2 answers
464 views
A prototypical problem for transfer matrix calculations in combinatorics
Let $G$ be a directed graph with self-loops on the positive integers: for every $n$, $G$ has the directed edges $(n,n)$ and $(n, n+1)$; additionally, if $n$ is even, $G$ has the directed edge $(n, n/2)...
- 9,721
5 votes
1 answer
234 views
Independence of parameter for eigenvalues of periodic family of tridiagonal matrices
Consider the family of matrices $C(\ell,\theta)\in \mathbb{R}^{2\ell+2\times2\ell+2}$, where $\ell\in\mathbb{N}$ and $\theta\in\mathbb{R}$ given by \begin{equation*} C(\ell,\theta)=\begin{pmatrix} ...
- 153
0 votes
1 answer
123 views
Extending totally unimodular matrices by a column
Given a matrix $M$, we will refer to the submatrix formed by the first $k$ rows as $M([k], \cdot)$. Let $A$ be a $m\times n$ totally unimodular matrix where $m \leq n$. We define a new $m\times (n+1)$ ...
- 123
4 votes
1 answer
159 views
Fastest way to compute Cesàro limit of the powers of a stochastic matrix
Let $P$ be a (finite) stochastic matrix. Let $$ C = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} P^k $$ be the Cesàro limit of the powers of $P$. What is the fastest known way to compute $C$?
- 2,440
0 votes
1 answer
342 views
Prove that a matrix is almost surely full rank
[Cross-posted from MS after 8 days without reply.] I have a real matrix $R_{ij} \in \mathbb{R}^{n \times m}$ whose entries are sampled iid from an absolutely continuous distribution $D$; a fixed real ...
- 155
5 votes
1 answer
223 views
Presentation of the algebraic closure of finite fields via matrices
Sorry if this question is too elementary for MO. Let $p$ be a prime and $F_p$ a field with $p$ elements and $F_{p^n}$ the field with $p^n$ elements Then we can choose an irreducible factor $f$ of ...
- 27.9k
0 votes
2 answers
273 views
Factorisation of bilinear polynomial over the integers
Let $f(x, y) = axy + bx + cy + d$ be a polynomial with integer coefficients $a, b, c, d$. Is there a criterion for $f$ to be factorised as $$ f(x, y) = (rx + s) (my + n) $$ for some integers $r, s, m, ...
- 49
6 votes
1 answer
353 views
A generalized Vandermonde matrix and spanning property
It is known that if $x_1, x_2, ..., x_n$ are all positive distinct real numbers, then the matrix $$ \begin{pmatrix} x_1^{a_1} & x_1^{a_2} & \cdots & x_1^{a_n} \\ x_2^{a_1} &...
- 113
0 votes
0 answers
142 views
Can it be proved that the specific real symmetric matrix is positive definite? (Numerically confirmed)
${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as $$\...
- 75
4 votes
2 answers
418 views
Expansion of $(A+t B)^{-1}$ as $t\to0$
Let $A, B$ be real matrices, with $A$ symmetric, positive semi-definite, with kernel spanned by the vector full of ones, and $B$ a non-singular matrix (we do not assume that $A$ and $B$ commute). Can ...
- 143
6 votes
1 answer
478 views
Numerical tests show the smallest eigenvalue of a certain matrix remains invariant when some parameters vary – any proof ideas?
I've numerically verified that the smallest eigenvalue of a specific matrix remains invariant when some parameters change. Looking for theoretical proof approaches. Appreciate any insights. ${\bf A} \...
- 75
2 votes
2 answers
335 views
Adjacency graphs and eigenvalues
I am currently reading this paper and this related paper (can also be found here), which explore the connection between Jordan normal forms and adjacency graphs. Theorem 6 in the first paper reads ...
- 79
0 votes
0 answers
134 views
Constraints on building adjacency matrices
We know that all adjacency matrices are square matrices. When our weighted directed graphs have loops or parallel edges, we obtain adjacency matrices that are, in general, asymmetric or non-Hermitian ...
- 79
0 votes
0 answers
59 views
Triangle of integer coefficients that contain both A113340 and A113350
Let $P(n,k)$ be A113340 (i.e., triangle $P$, read by rows, such that $P^2$ transforms column $k$ of $P$ into column $k+1$ of $P$, so that column $k$ of $P$ equals column $0$ of $P^{2k+1}$, where $P^2$...
user231922
0 votes
0 answers
109 views
Quadratic equation involving diagonals of inverses of matrices
Let $b$ and $c$ be two real numbers, $D \in \mathbb{R}^{n \times n}$ be a diagonal matrix, $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, $Q \in \mathbb{R}^{n \times n}$ be a symmetric matrix ...
- 11
9 votes
4 answers
563 views
How many translates of the singular‐matrix hypersurface are needed to cover $M_n(\mathbb{F}_2)$?
Let $n$ be a positive integer, and consider the hypersurface of singular $n\times n$ matrices over $\mathbb{F}_2$, denoted $$ \mathcal{S}_n = \{X\in M_n(\mathbb{F}_2) : \det(X)=0\}. $$ Note that \...
- 469
0 votes
1 answer
242 views
Is this formula for a matrix block inverse in terms of the entire matrix inverse known? [closed]
I needed such a formula and when I couldn't find it on Wikipedia, I asked Claude.AI to help me derive one and this is what we came up with: The formula: Given an invertible matrix partitioned as $$P = ...
- 175
2 votes
0 answers
111 views
Is it possible to analyze the eigenvalue of a specific tridiagonal matrix?
I'm considering the $n \times n$ tridiagonal matrix $$ A = \begin{pmatrix} 0 & 1 & & & \\ 1 & c & 1 & & \\ ...
- 21
6 votes
0 answers
110 views
Concentration bound for the number of full-rank matrices in a random subspace of matrices
Let $\mathcal{M}_{n,m}(\mathbb{F_q})$ be the set of $n$ by $m$ matrices over the finite field of order $q$, with $n \geq m$. For a (non-trivial) subspace $V \subset \mathcal{M}_{n,m}(\mathbb{F_q})$, ...
- 271
2 votes
1 answer
160 views
Trace maximization for products of symmetric nonnegative matrices summing to $J_n$
Let $n$ be a positive integer and $A_1, A_2, \dots, A_k$ be a sequence of real symmetric $n \times n$ matrices with nonnegative entries, such that $$A_1 + A_2 + \dots + A_k = J_n,$$ where $J_n$ ...
- 613
0 votes
0 answers
35 views
How to estimate the tail probability of a structured Frobenius norm involving Gaussian noise?
Let $ Q \in \mathbb{R}^{n \times m} $ be a known (non-random) invertible matrix, and let $ W \in \mathbb{R}^{m \times d} $ be a random matrix whose entries are i.i.d. Gaussian variables: $ W_{kj} \sim ...
- 1
18 votes
2 answers
488 views
Number fields in fast matrix multiplication
A common approach to construct fast multiplication algorithms is to make an ansatz for the matrix multiplication tensor of fixed dimension and rank (e.g. $2 \times 2 \times 2$ and rank $7$ if we want ...
- 2,602
1 vote
0 answers
131 views
On the reducibility of a characteristic polynomial
Given $n \times n$ Hermitian matrices $A$ and $B$, we define the polynomial, $$g(\alpha, \beta) := \det(A + \alpha B - \beta I),$$ where $I$ is the $n \times n$ identity matrix. It may be seen as the ...
2 votes
1 answer
291 views
Null space of infinite-dimensional matrix
Is there a general way to tell if the null space of an infinite-dimensional matrix contains a vector that is not zero? This question connects to the following problem: If I know that \begin{equation} \...
- 71
3 votes
0 answers
80 views
Variational problem of minimizing sum of Frobenius norms
I'm stumped by the following variational problem which came up in the course of my research. Let $X_1, X_2 \in \mathbb{R}^{m \times d}$ and $Y_1, Y_2 \in \mathbb{R}^{n \times d}$ be fixed matrices of ...
- 1,901
2 votes
1 answer
194 views
Difference sets, skew-Hadamard matrices and elements in group rings
Let $G$ be a finite group of order $v = 4n - 1$, and let $D \subset G$ be a subset such that: $1 \notin D$, $G$ is the disjoint union of $D$, $D^{-1}$, and $\{1\}$, where $D^{-1} = \{ d^{-1} \mid d \...
- 123
15 votes
1 answer
1k views
Diagonalizing Pascal's triangle
Let $D_n$ be the $n \times n$ diagonal matrix with entries $1, 2, \dots, n$. Let $P_n$ be the $n \times n$ upper triangular matrix whose entry $a_{i,i+j}$ is given by $\binom{i+j}{i-1}$. For instance, ...
- 10k
2 votes
1 answer
89 views
Decomposition of a matrix relative to a diagonal matrix: how to prove $\tilde M=\tilde \Omega +\tilde B +\tilde N \Lambda^{-1}$?
Let $\Lambda$ be a non-degenerate $n \times n$ diagonal matrix with distinct non-zero entries. It is known (see Constitutive laws for the matrix-logarithm of the conformation tensor by Fattal and ...
- 21
1 vote
1 answer
130 views
Inequalities for Inverses of Strictly Diagonally Dominant Matrices
We consider a symmetric matrix $X$ of size $n \times n$ that is strictly diagonally dominant, i.e. $X_{ii} > \sum_{j \neq i} |X_{ij}|$ with strictly positive diagonal terms $X_{ii} > 0$ and ...
- 2,602
16 votes
1 answer
1k views
Matrices with many traceless powers
Suppose that a nonsingular real $3\times3$ matrix $A$ is such that the trace of $A^k$ vanishes for infinitely many positive integers $k$. Must $A^m$ be a scalar matrix for some $m$? The problem ...
- 4,236
10 votes
2 answers
526 views
Two square matrices $A, B$ such that $A^k, B^k$ have identical diagonals for all $k$
[Edited in light of helpful comments below] What can be said about two matrices $A,B\in M_n(\mathbb R)$ such that $A^k$ and $B^k$ have identical main diagonals for all $k\in\mathbb N$? Some more ...
- 863
1 vote
0 answers
166 views
Matching matrix columns under scaling, translation and orthogonal transformation
Given vectors ${\bf p}_1, {\bf p}_2, \dots, {\bf p}_n \in {\Bbb R}^d$ and ${\bf q}_1, {\bf q}_2, \dots, {\bf q}_n \in {\Bbb R}^d$, define the $d \times n$ matrices $$ {\bf P} := \begin{bmatrix} ...
- 11
11 votes
2 answers
565 views
Preperiod of powers of matrices modulo m
Let $A$ be a square matrix with integer entries and let $m$ be a positive integer. From the pigeonhole principle it follows easily that the sequence $$I,A, A^2, A^3,\; \dots \pmod m$$ is eventually ...
- 113
5 votes
1 answer
394 views
Characterization of a convex sum of determinants
A quantum information problem I have been thinking about comes down to a linear algebra question that I dare to ask here. Given: Integers $N,P$, and a set of real positive coefficients $C_1,C_2,\ldots ...
- 200k