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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15
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727 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
402 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 ...
12
votes
0answers
333 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 ...
12
votes
0answers
272 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 - ...
12
votes
0answers
699 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 ...
11
votes
0answers
565 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 ...
9
votes
0answers
456 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
155 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
266 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
447 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
294 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
202 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
151 views

LU factorization for $I+A$ (A skew-symmetric)

The matrix $M=I+A\in \mathbb{R}^{n\times n}$, where $I$ is the identity and $A=-A^T$ is skew-symmetric satisfies $$ Mx\cdot x = \|x\|^2 \quad\text{for all }x\in\mathbb{R}^n. $$ Therefore, $M=LU$ has ...
7
votes
0answers
173 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
320 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
600 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
240 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
195 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
87 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
158 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
167 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
331 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
196 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
518 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
390 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
287 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: ...
4
votes
0answers
81 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
103 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
163 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
101 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
235 views

How to calculate the inverse of a matrix with block Kronecker structure?

I have a Hermitian matrix $A$ with the following block Kronecker structure, $A = \begin{pmatrix} R_0 \bigotimes S_0 & R_1 \bigotimes S_1\\ R_1^H \bigotimes S_1^H & R_0 \bigotimes S_0 ...
4
votes
0answers
140 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
351 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
141 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
428 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
817 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
543 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
408 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
268 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
220 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 $\lbrace -N,-N+1,\dots,N-1,N\rbrace$. ...
4
votes
0answers
1k views

Matrix normalization for constant row and column sums

From a mxn matrix M with only strictly positive elements, I want to multiply each row and each column by real values in order to obtain a matrix N whose row sums are all 1/m and column sums are all ...
4
votes
0answers
240 views

A matrix minimisation problem

Feel free to edit the title! Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices. Question: If there are $t\in\mathbb R$ and ...
4
votes
0answers
355 views

The order of the Jacobi method for Hermitian matrices

Let $H$ be an $n\times n$ Hermitian matrix. The Jacobi method is an iterative method for finding the spectrum of $H$. It is described in every book on numerical linear algebra. Principle: At step ...
4
votes
0answers
421 views

Haar measure on strictly upper triangular matrices

Let F be a function field, and A its adele ring. I want to consider U(A)/U(F), where U(A) is the space of strictly upper triangular matrices with entries from A, and U(F) is the same with entries ...
4
votes
0answers
473 views

An inverse eigenvalue problem on Jacobi matrices

I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
3
votes
0answers
117 views

Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
3
votes
0answers
92 views

Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem. Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that $$ \| D_1 A ...
3
votes
0answers
206 views

The Poisson-kernel in the plane and polynomials

Let \begin{align*} p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\ & = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j} \end{align*} be a non-constant complex polynomial with ...
3
votes
0answers
90 views

Inverting a matrix with entries equal to positive or negative infinity

I would like to define an inverse on matrices whose entries may be positive or negative infinity. To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
3
votes
0answers
82 views

Large prime divisors in entries of matrix powers

Are any examples known of an integer matrix $A$, such that the largest prime divisor of some specified entry of $A^n$ grows exponentially in $n$? How about where it just grows strictly faster than any ...