# 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.

2,061 questions
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### The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up: Let us consider the space of all $m \times n$ real matrices, and define a scalar ...
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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 $mn$...
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### Square root of a large sparse symmetric positive definite matrix

I am trying to calculate $$Y = A^{\frac 12} X$$ where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
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### transfer a relation from matrices to their cholesky decompositions

I'm stuck on a seemingly easy problem. $$LL' = TAA'T'$$ a matrix A (lower triangual matrix -> derives from a cholesky decomposition of a covariance matrix -> ( A * A' ) is positive semidefinite and ...
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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 ...
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### Solving a “reversed” Stein equation

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{\star} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}.$$ My question. Is it true ...
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### Convexity of the matrix mapping $X^{-2}$

Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex? Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
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### On the (qualitative) behavior of a coupled differential equation

Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following ...
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### Rotatable matrix, its eigenvalues and eigenvectors

We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change. I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
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### Representation of a matrix ring

Years ago, I read a paper about how to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequences of two simpler matrices $S$ ...
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### On the basis of a finite dimensional vector space (revised)

Revision in response to the comments to earlier version: To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
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### Diagonal elements of Hermitian matrices with given eigenvalues

Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
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### Depth bound of generating an matrix algebra

Let $X$ be Hilbert spaces $\mathbb{C}^d$, and $L(X)$ be the sets of linear operators of $X$. We are given a matrix subspace $S\subset L(X)$. Via the following procedure, one can generate the smallest ...
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### L1 norm constraint on product of 2 matrix

I want to solve below minimization problem \begin{equation*} \begin{aligned} & \underset{A, B}{\text{minimize}} & & ||Y-AB^T -D||_F^2 \\ & \text{subject to} && |A_i|_1 \leq a,...
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### Solution of $\dfrac{d [\epsilon_i]}{dt} = (\beta \mathbf{A} -\delta \mathbf{I})\left[\epsilon_i\right] - \alpha \mathbf{B} \left[\epsilon_i^2\right]$

I have a problem that can be described by the following equation: \dfrac{d \left[\epsilon_i\right]}{dt} = \left( \beta \mathbf{A} - \delta \mathbf{I} \right) \left[\epsilon_i\right] -...
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### higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
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### identity involving spectral functions

Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true : $$A^* f(AA^*) = f(A^* A) A^*$$
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### Symmetric Grothendieck inequality

Grothendieck's inequality states that for all $n \times n$ matrices $(a_{ij})$ such that $$\max_{x \in \{\pm 1\}^n,\, y \in \{\pm 1\}^n} \left|\sum_{ij} a_{ij}\, x_i\, y_j\right| \leq 1,$$ there ...
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### An inequality regarding projection

Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define ...
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### Upper bound on the number of non-zero entries of the product of sparse matrices

I have two sparse matrices: $A$ of dimension $m \times k$ and $B$ of dimension $k \times n$. Is there a way to know how many non-zero entries there are in $C = A B$ without computing $A B$? I can ...
This was asked earlier at MSE and incorporates replies from Omnomnomnom and Chappers. Let A = (a$_{i,j}$) be an $\,$ n x n $\,$ real symmetric matrix. What can be said about lower bounds for rank(...
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let \$\...