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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18
votes
0answers
478 views

Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 ...
17
votes
0answers
825 views

The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
15
votes
0answers
444 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
13
votes
0answers
709 views

Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
12
votes
0answers
300 views

Descartes rule of signs for a noncommutative polynomial in matrix variables

Recently, while studying certain notions of "averaging" a set of input matrices, I obtained a nonlinear polynomial in matrix variables. A simple example is \begin{equation*} \mathcal{G}(X) := X^n - ...
11
votes
0answers
219 views

Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on. I am aware that even Gantmacher 1959 has this terminology however I don't know ...
11
votes
0answers
234 views

Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - ...
11
votes
0answers
579 views

Regular languages of matrices and their generating functions

My question is somewhat related to this question. Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
10
votes
0answers
239 views

Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
10
votes
0answers
493 views

Is “being a full ring of quotients” a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
8
votes
0answers
164 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using >this< formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= ...
8
votes
0answers
319 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
8
votes
0answers
476 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...
8
votes
0answers
300 views

What is the “positive part” of the unit ball in $M_n(R)$ ?

In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm $$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$ where $\|x\|$ is the Euclidian norm. The closed unit ball $B$ is the set of contractions (in the ...
8
votes
0answers
215 views

Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial

This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But.... Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
7
votes
0answers
182 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
0answers
179 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
6
votes
0answers
343 views

Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...
6
votes
0answers
625 views

inverse eigenvalue problem on graph laplacian

I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody ...
5
votes
0answers
118 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb R^{n\times n}$, is there a nice method to characterize matrices $Q\in\Bbb R^{n\times n}$ such that $$\mathsf{rank}(M+Q)\leq s\cdot\mathsf{rank}(Q)$$with some fixed $s>0$? ...
5
votes
0answers
57 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
5
votes
0answers
193 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
5
votes
0answers
260 views

Transforming a binary matrix into triangular form using permutation matrices

I am interested in the complexity of the following problem: Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix? I am also interested in ...
5
votes
0answers
123 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
5
votes
0answers
543 views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
5
votes
0answers
266 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 ...
5
votes
0answers
202 views

Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for ...
5
votes
0answers
88 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
5
votes
0answers
161 views

Bound on number of multiplications required to generate a matrix algebra from generators?

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
5
votes
0answers
172 views

Largest entry of the inverse matrix?

I wonder if there is a "qualitative way" of predicting from the structure ix of the matrix $A$ which entry of $A^{-1}$ will be the largest. I am specially interested in the case that $A$ is a ...
5
votes
0answers
355 views

Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
5
votes
0answers
215 views

concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
5
votes
0answers
576 views

Status of Hadamard matrix conjecture

I would like to know if any progress has been made on Hadamard conjecture : There exists a Hadamard matrix of order $n=4k$ $\forall k \in \mathbb{N}$.
5
votes
0answers
396 views

symplectic matrices

If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that $\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$? This arose in connection ...
4
votes
0answers
77 views

Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries). Can we partition this union into at most $n$ rectangles? I think it's pretty ...
4
votes
0answers
84 views

Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?

--Updated description-- I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test. Suppose there is a vector space ...
4
votes
0answers
39 views

Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup

I tried asking this on math exchange, but no luck, so thought I'd try here. Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
4
votes
0answers
89 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 ...
4
votes
0answers
111 views

Concept of eigenvector restricted to nonnegative entries

Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem $\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
4
votes
0answers
207 views

How to find the unitary matrices in this exponential matrix representation

In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials: if $A$ and ...
4
votes
0answers
125 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
4
votes
0answers
149 views

Examples of functions from matrices to real numbers with certain properties

Let $M(\mathbb{R})$ be the set of all matrices (of any size) over $\mathbb{R}$. Let $v : M(\mathbb{R}) \rightarrow \mathbb{R}$ be a function which satisfies the following 5 properties: If ...
4
votes
0answers
404 views

A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE): $\frac{d}{dt}y= A(t) y$, where $A(t)$ is a ...
4
votes
0answers
146 views

connectivity in automata by words of length n-1

Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected? That is for any pair of distinct ...
4
votes
0answers
455 views

Characteristic polynomial of a symmetric integer matrix

I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a ...
4
votes
0answers
855 views

Cartan decomposition for upper triangular matrices

Due to the comments, I have the impression that I have to be more precise. Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$. Let $K= GL_n(o)$ and let $I$ the Iwahori ...
4
votes
0answers
550 views

Path connected set of matrices?

Consider the collection of $n$ by $n$ matrices $$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$ where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint ...
4
votes
0answers
412 views

Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
4
votes
0answers
275 views

What are the eigenvectors of the Lagrange interpolation matrix?

Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - ...
4
votes
0answers
227 views

Growth of symmetric positive definite integral matrices.

Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$. ...