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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Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle of $G=\GL_{n,\R}\,$, that is, an invertible ...
Mikhail Borovoi's user avatar
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Spectrum of large Hilbert matrices

Let $x_k>0$ be a increasing sequence of real numbers, such that $$\sum_0^\infty\frac1{x_k}<+\infty.$$ Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with $$a_{...
Denis Serre's user avatar
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Estimates of the Frobenius norm of commutator

Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
BharatRam's user avatar
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What is the minimum nonzero rank in a random subspace of matrices?

Fix positive integers $m$, $n$, and $k\leq mn$, and draw a $k$-dimensional subspace $S\leq\mathbb{R}^{m\times n}$ uniformly from the Grassmannian. What is known about the random variable $R(m,n,k):=\...
Dustin G. Mixon's user avatar
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A question on eigenvalue of parametric matrix

Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
DSM's user avatar
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Graph-class defined by matrix-like vertex-operations

Let $m$ be a positive integer. We define a (directed) graph on $m(m-1)$ vertices $$V = \bigl\{(i,j) \mid i \ne j,\, i,j\in\{1,\dots,m\}\bigr\}$$ and edges as follows: $(i,j) \in V$ is adjacent (...
Daniel Krenn's user avatar
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Exponential of infinite dimensional matrix

Originally posted on Math SE but didn't get any responses. Thus, I thought I would ask here with some more details. I have a matrix originating from Master Equation for birth death process on semi ...
plambda's user avatar
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Zariski density for certain subsemigroups

$\DeclareMathOperator\GL{GL}$Suppose that we have a Zariski-dense subgroup $\Gamma$ of $\GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series $$ \sum_{x \in \Gamma} e^{-s \log\|...
Zestylemonzi's user avatar
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Minimal set generators ideal submaximal minors

Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as: $$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
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Color algebras and color involutions

If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined ...
Fabrizio's user avatar
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Reference for matrices with all eigenvalues 1 or -1

In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues ...
Mare's user avatar
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Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE. Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
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Is there a $3\times 3$ matrix over a Dedekind domain not similar to a matrix with zero top right entry?

Let $R=\mathbb{Z}[\sqrt{-5}]$, which is well known to be a Dedekind domain but not a PID. Let $\mathrm{M}_{3}(R)$ be the set of $3\times3$ matrices over $R$. Does there exist a matrix $A\in\mathrm{M}_{...
A Stasinski's user avatar
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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 $n \times n$ ...
Nathaniel Johnston's user avatar
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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}\...
Artemy's user avatar
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Matrix series with Hadamard products

Let $A$ and $B$ be hermitian matrices (a special case that would already help would be $A^{-1} = B^T$). I'm looking for a closed form of the series $$X := \sum_{n=0}^\infty A^n \circ B^n$$ where $\...
alcubierre-drive's user avatar
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Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?

Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
Ludwig's user avatar
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Rank of binary matrix related to the number of positive squarefree integers less than $n$

I posted this question at the Mathematics SE, but received no response there so I am posting it here. The following fact is stated in the comments-section of sequence A013928 in the OEIS. Let $C$ ...
Pietro Paparella's user avatar
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How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ $$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
Maryam Hak's user avatar
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Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
wonderich's user avatar
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Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

I am ready to agree beforehand that this looks more like a math.SE question. I posted it there a week ago without any feedback (except for 27 views and 2 upvotes). Besides, I really need an answer. ...
მამუკა ჯიბლაძე's user avatar
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Totally Unimodular matrix edited from ordinary matrix

Given a matrix $M\in\{0,1\}^{m\times n}$ is there an algorithm to tell if we can convert some of $1$s to $-1$s and make $M$ Totally Unimodular and output such a Totally Unimodular in polynomial in $mn$...
Turbo's user avatar
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An upper bound on the Jordan condition number of a matrix

The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
Daniel86's user avatar
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Maximizing a certain eigenvalue ratio

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
Ludwig's user avatar
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A Toeplitz variant of the Hilbert matrix

It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries $$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$ determines a bounded operator on $\ell^{2}(\mathbb{N}...
Twi's user avatar
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Distribution of min/max row sum of matrix with i.i.d. uniform random variables

Given a $n\times n$ symmetric random matrix such that all diagonal elements are all fixed as $1$. all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
Tony's user avatar
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0 answers
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Non-singularity of a series of matrices

Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two ...
Ludwig's user avatar
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"Singularly convex" cones of matrices

The ambient space if ${\bf M}_n({\mathbb R})$. Let us begin with facts. 1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
Denis Serre's user avatar
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix

(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows: $$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$ Let $...
Kazuki OKAMURA's user avatar
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0 answers
218 views

Extending a Theorem of Brualdi to Matrices with Infinitely Many Rows

This question is about extending a result on transportation polytopes from Brualdi regarding $m\times n$ matrices to the case when $m=\infty$. Notation: Denote an $m\times n$ matrix by $A=[a_{i,j}]$, ...
The Substitute's user avatar
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0 answers
143 views

Zero diagonal nonsymmetric block checkerboard matrix: orbits and numerical ranges

Let $A \in \mathbb{R}^{m \times m}$ be a nonsymmetric zero diagonal matrix with a zero/non-zero pattern which is symmetric and persymmetric (i.e. symmetric in the northeast-to-southwest diagonal). If ...
Astor's user avatar
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0 answers
181 views

Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
Henry.L's user avatar
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Is there a nice way to express a matrix exponential when rows are proportionally scaled?

Assume I am given an $n \times n$ matrix $A$ with real or complex coefficients. Its matrix exponential is denoted by $\exp(A)$ and is calculated as usual. Assume further that I want to rescale the ...
tobias's user avatar
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What is the time complexity of the largest singular value and its vectors?

Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
B. Arsic's user avatar
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0 answers
80 views

Perturbation of a rank-restricted product of matrices

I formulated a statement, which is hopefully true (at least I'm not knowledgeable enough to see a reason for it not to be). However, I'm struggling to come up with a proof. Let $W_i \in \mathbb{R}^{...
Serghei's user avatar
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Bessel in matrix?

Let $M_n$ be the matrix $$M_n=\begin{pmatrix} 1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\ 1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
T. Amdeberhan's user avatar
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93 views

Are extremal tournament matrices always circulant or 'almost circulant'?

Define an antisymmetric 1-x-matrix as an $n\times n$ matrix $M=(m_{ij})$ with $m_{ii}=0$ and $\{m_{ij},m_{ji}\}=\{1,x\}$ for all $1\le i<j\le n$. Call their set $\mathcal A_n$. The setup is as ...
Wolfgang's user avatar
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0 answers
560 views

Determining whether a Schur complement is invertible

Consider the symmetric matrix $$M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix}$$ where $A \in \cal{R}^{n \times n}$ and $C \in \cal{R}^{m\times m}$ are symmetric, ...
berkin's user avatar
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0 answers
235 views

p-adic analogue of self-adjoint operator

Consider the very well-known result that any Hermitian matrix over $\mathbb{C}$, say $T$, admits a decomposition $T = UDU^*$ where $U$ is unitary and $D$ is diagonal with real entries. I am looking ...
GiantTortoise1729's user avatar
4 votes
0 answers
112 views

Inducing surjections on $GL_n(-)$?

Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
BillScroggs's user avatar
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0 answers
212 views

Equations in finite subgroups of unitary groups

Let $n$ be an integer. Andreas Thom mentioned that Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ ...
Bjørn Kjos-Hanssen's user avatar
4 votes
0 answers
433 views

A sum of Ramanujan sums

I have the following question about Ramanujan sums. (All vectors and matrices here will be understood to have integer entries.) Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...
tomos's user avatar
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4 votes
0 answers
147 views

Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring

Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$. For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
stupid_question_bot's user avatar
4 votes
0 answers
83 views

Matrices with almost constant coefficient have a simple eigenvalue

As a by-product of a general result for bounded operators of a Banach space, I have the following: A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
Benoît Kloeckner's user avatar
4 votes
0 answers
167 views

Fast matrix-vector product for structured matrices

Let $X\in\mathbb{C}^{m\times n}$ be a matrix that satisfies the Sylvester equation $$AX-XB = F,\qquad A\in\mathbb{C}^{m\times m}, \quad B\in\mathbb{C}^{n\times n},$$ where $F\in\mathbb{C}^{m\times n}$...
Diego Ruiz's user avatar
4 votes
0 answers
368 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e., the matrix is not only weak diagonal-dominant, but ...
Skrodde's user avatar
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0 answers
207 views

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
user173856's user avatar
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4 votes
0 answers
123 views

How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$, and it is finitely generated by some known generators. That is, $G=\langle g_1,\dots,g_n\rangle$. The Frobenius norm of a matrix $m=\...
j0equ1nn's user avatar
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4 votes
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468 views

Efficiently calculate the trace of the product of two large but symmetric matrices, one of which is an inverse

Sorry about the long title. I need to calculate the trace of $M(M+D)^{-1}$, where $M$ is a dense symmetric matrix, and $D$ is a diagonal matrix. The main issue is the dimension could be large (usually ...
aenima's user avatar
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4 votes
0 answers
401 views

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
user87933's user avatar

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