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12 votes
0 answers
825 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 ...
Wolfgang's user avatar
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4 votes
0 answers
196 views

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
3 votes
0 answers
420 views

(Expected) Size of smallest singular value of a Vandermonde matrix associated to roots of polynomial

Let $n,H$ two fixed positive integers. Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...
user70925's user avatar
  • 313
3 votes
0 answers
182 views

Spectral radius of infinite substochastic upper triangular matrix

Let $M$ be a Markov chain on $\{0, 1, 2, \dots\} \cup \{\delta\}$, where $\Pr(i \to j) > 0$ for $i, j \in \mathbb{N}$ only if $j > i$, and $\Pr(\delta \to \delta) = 1$. This represents a birth-...
Kevin's user avatar
  • 131
3 votes
0 answers
549 views

Canonical forms for block-positive-definite matrices

Suppose we are given a block $2\times 2$ matrix that is positive-definite, and let's suppose for simplicity that the blocks along the main diagonal are the identity. So $$ \begin{bmatrix} I & X \\\...
Laurent Lessard's user avatar
2 votes
0 answers
181 views

Is every nearly rank-1 doubly stochastic matrix a product of pairwise averaging matrices?

A doubly stochastic matrix is a square matrix with non-negative real entries where the sum of each row is $1$ and the sum of each column is $1$. A pairwise averaging matrix is a matrix of the form $tA+...
Joseph Van Name's user avatar
2 votes
0 answers
326 views

Explicit formula for this distance between positive semi-definite matrices?

Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I ...
Lénaïc Chizat's user avatar
2 votes
0 answers
102 views

Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries

I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
Ian Cero's user avatar
  • 121
0 votes
0 answers
66 views

Random elliptical potential lemma

Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
Mixi Andrew's user avatar
0 votes
0 answers
45 views

On full rank submatrices of a construction

Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
VS.'s user avatar
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0 votes
0 answers
47 views

"Probability" for a partitioned matrix to be singular

Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix $$ M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
Ludwig's user avatar
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