# 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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### Maximum rank in a class of $0\,$-$1$ partitioned matrices satisfying combinatorial constraints

We are given a matrix $M \in \{0,1\}^{n\times n}$ satisfying the following property. The rows and columns of $M$ can be partitioned into $k$ rowgroups and $k$ colgroups respectively, such that in ...
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### How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently? [closed]

I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where $y$ is a $n \times 1$ vector. $A$ is a $m \times n$ matrix where $n \gg m$. $B$ is a $m \times m$ symmetric positive definite matrix; the ...
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### Non singularity of a generalised Vandermonde matrix through Hadamard product

I'm currently trying to prove the following. Consider $k_1,\dots,k_N$ complex numbers not lying on the real and imaginary axes. Then consider W_N(x)= \text{Wronskian}\big(\cosh(k_1x),...
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### Proving that a matrix is positive semidefinite

Let matrices $A, B$ be positive semidefinite. Can we prove that $A(I+BA)^{-1}$ is positive semidefinite?
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### Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: http://www.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
Let $A \in \mathbb R^{n\times n}$ and assume that all eigenvalues lie in the left open halfplane. Is it true that the logarithmic norm $\mu_2 (A):= \lambda_{\max} \left(\frac{A + A^T}{2}\right)<0?... 1answer 298 views ### Gaussian integrals over the space of symmetric matrices Let$S\in\mathcal S_N$be a$N\times N$symmetric matrix over the reals, and introduce the (normalised) gaussian measure $$\mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\... 2answers 540 views ### Inverse of particular lower triangular matrix I have an n \times n lower triangular matrix A where$$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>jA_{ii}=1, \quad 1 \leq i \leq n,$$and {\bf x}_i is a 1 \times k (row) vector, where ... 0answers 59 views ### Maximum number of negative entries in a matrix with positive diagonal and given rank Suppose A \in \mathbb{R}^{n \times n} has positive entries on it main diagonal and \mbox{rank}(A) =: d < n. Then, what is the maximum number of many negative entries A can contain? If, in ... 0answers 78 views ### A Toeplitz variant of the Hilbert matrix It is well-known that the Hilbert matrix H, i.e., the symmetric Hankel matrix with entries$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$determines a bounded operator on$\ell^{2}(\mathbb{N}...
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form \begin{gather} \begin{pmatrix} ...