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Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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Moments on the Stiefel manifold

Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
Drew Brady's user avatar
3 votes
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Matrix-Gaussian distributions

The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
user3826143's user avatar
2 votes
1 answer
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"High complexity" of eigenbasis of Wigner matrices?

Let $W$ be an $N \times N$ complex Wigner matrix, i.e. i.i.d. entries restricted to Hermitian matrices. Let $W=UDU^{\ast}$, i.e. $U$ encodes the eigenbasis of $W$. Are there any statements known about ...
Marcin Kotowski's user avatar
2 votes
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Spectral bound for sample covariance matrix without assuming $X = \Sigma^{\frac{1}{2}} Z$

Let $X$ be a random $(p \times n)$-matrix with iid centered columns and suppose the entries of $X$ all have light tails (in a strong enough sense, for example sub-Gaussian). Are there any results ...
Tardis's user avatar
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What is known about the distribution of eigenvectors for random positive semidefinite matrices?

Let $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$. Define the random $d \times d$ real positive semidefinite matrix, $$ S_n = \frac{1}{n} \sum_{i=1}^n ...
Drew Brady's user avatar
1 vote
1 answer
253 views

Does free multiplicative convolution become free additive convolution under logarithm?

Let $X$ and $Y$ be two $n\times n$ random matrices that have zero measure over degenerate matrices. For a positive definite matrix with eigen-decomposition $A = U \Lambda U^\top$, let $\log(A) = U \...
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Bound p-th order moments for a random Wishart matrix to show the sub-exponential property

Let $a\in\mathbb{R}^k$ be a random vector sampled from $N(0,\Sigma)$. Let $X = aa^T - \Sigma$. Then we have $\mathbb{E} X = 0$. Can we find a constant $C\in\mathbb{R}$ and another fixed matrix $A\in\...
Nicole's user avatar
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1 answer
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Matrix concentration inequality for unbounded (sub-exponential) matrices

Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...
Nicole's user avatar
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7 votes
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Wishart matrices: are eigenvalues and eigenvectors independent?

Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries. In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
Drew Brady's user avatar
1 vote
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Limiting value of Stieltjes transform of sum of independent Wishart matrices

Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
dohmatob's user avatar
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2 votes
2 answers
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Behavior of a Wishart quadratic form

Let $X \in \mathbb{R}^{n \times d}$ be a random matrix with iid standard Gaussian entries. Let $e_1$ denote the first canonical basis vector in $\mathbb{R}^d$. Define $$ P_d(\lambda) = (1-\lambda) e_1 ...
Drew Brady's user avatar
2 votes
1 answer
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Orthogonal projection $X X^+$ from random Gaussian matrix $X$

Given a standard Gaussian matrix $X\in\mathbb{R}^{n\times d}$, $d<n$, with entries sampled i.i.d. from $\mathcal{N}(0,1)$, is the corresponding orthogonal projection $X X^+ = X (X^\top X)^{-1} X^\...
João F. Doriguello's user avatar
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1 answer
226 views

Inequality with Hermite polynomials

Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by $$\sqrt{\sqrt{\pi} 2^n n!}$$ for the purpose of normalization. These are orthogonal with respect to the weight function $e^{...
T. Amdeberhan's user avatar
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Dot product of a randomly orientated vector and a fixed vector

Let us consider a random variable $Z$ with a probability density function $f$ with respect to the Haar measure on $\mathrm{SO}(3)$. Next, we consider two fixed normal vectors $u,v$ in $\mathbb{R}^3$. ...
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RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)

Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
dohmatob's user avatar
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Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?

I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors. Exercise 3.1.7 from the book is the following. Let $X =...
user135520's user avatar
2 votes
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Random matrix with power law decay in eigenvalues

What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact. I want to find random matrix ensembles that naturally ...
CWC's user avatar
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6 votes
1 answer
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Poisson kernel for the orthogonal groups

For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
thedude's user avatar
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Random pseudo-inverse matrix problem

Given a matrix $M \in \mathbb{R}^{n \times N_d}$, $N_d \gg n$ and $\mathrm{rank}(M) = n$, the entries of $M$ are denoted as $M_{[ij]}, i = 1,...,n, j = 1,..., N_d$ and $M_{[ij]} \in [-\textbf{m}, \...
Michael Miller's user avatar
1 vote
1 answer
101 views

Bounded density for determinant of GOE

Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
kaleidoscop's user avatar
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1 vote
1 answer
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Maximum column norm of random $A^{-1}B$

Suppose that $A$ is an $n$ by $n$ Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let $b$ be a $n$-Gaussian vector. Then it could be easily proven that the ...
ZZZZZZ's user avatar
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Multilinear non-commutative Khintchine inequality

Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix. Question: Is there a known bound for ...
user293794's user avatar
3 votes
1 answer
408 views

Computing Haar measure of matrices sampled from SO(n)

I am looking to sample uniform matrices from SO(n). I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
magnesium's user avatar
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1 vote
2 answers
59 views

Distribution of the constraint matrix conditioned on the solution of the linear system

Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
ZZZZZZ's user avatar
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Question about the spectrum of a deformed GOE matrix

Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the ...
EJAV's user avatar
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Convergence of edge eigenvalues for Gaussian matrices

I am reading this lecture note. I have a difficulty in understanding the third section in chapter 6. Particularly, in Theorem 4.1, they claimed that Let $X$ be a Gaussian Wigner matrix satisfying ...
Pipnap's user avatar
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2 votes
1 answer
114 views

Functional calculation for Hermitian matrices

First, let me recall some useful definitions We recall that if $A=U \Lambda U^*$ is a Hermitian matrix with $U U^*=U^* U=I$ and $\Lambda=\operatorname{diag}\left(\lambda_1\right)$ and $f: \mathbb{R} \...
Pipnap's user avatar
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Convergence in probability of quadratic form with positive mean

Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
Student88's user avatar
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Limiting value of trace of resolvent matrix involving two independent Wishart random matrices

Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that $$ d/n_k \to \phi_k \in (0,\infty). $$ Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
dohmatob's user avatar
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Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?

This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
Ma Joad's user avatar
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2 votes
1 answer
121 views

Eigenvalue analysis of $X^T (XX^T + \mathrm{Id})^{-1} X$ for $X$ iid random matrix

Consider the following quantity $$X^T (XX^T + \mathrm{Id})^{-1} X,$$ where $X \in \mathbb{R}^{m\times n}$ is a iid random matrix with 0 mean and finite variance. The empiric covariance matrix ${X^T X}$...
Goulifet's user avatar
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Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?

Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function $$ f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big], $$ over the class of real ...
Drew Brady's user avatar
1 vote
2 answers
286 views

Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
6 votes
1 answer
165 views

A second-order recursion (functional equation)

In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion, $$L(s+1)=L(s)+2s(2s+1)L(s-1).$$ Is it familiar to someone ? Is ...
Denis Serre's user avatar
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Conditional distributions of random orthogonal projection matrix

I have encountered a rather curious question. Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...
Landon Carter's user avatar
3 votes
2 answers
211 views

Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$

Is there a closed-form solution for $$\max_D \operatorname{Tr}(ADBD)$$ where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
CWC's user avatar
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1 vote
0 answers
68 views

Moment method / genus expansion for random matrices with i.i.d. entries

Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$ ...
Panda Jonas's user avatar
1 vote
0 answers
65 views

Dimension-free sample complexity for the inverse of Gaussian sample covariance?

Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
axk's user avatar
  • 517
9 votes
2 answers
538 views

What are applications of asymptotic freeness of random matrices?

In around 1990 Voiculescu showed asymptotic freeness of certain random matrices, i.e., free independence when the matrix size goes to infinity. Since then this link between free probability and random ...
Bart's user avatar
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1 vote
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Random matrices: Relation between leading eigenvector and a vector in culumn space

Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
legon's user avatar
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236 views

$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder

I'd like to share a problem that I have been dealing with for a longer time now. In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
Fred Hucht's user avatar
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4 votes
1 answer
429 views

Reference book on Riemann zeta function and random matrices

What is a reference book to understand the relation between the Riemann zeta function and random matrices?
Cosimo's user avatar
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2 votes
1 answer
217 views

Expected norm of a product of Gaussian matrices

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof? $$E[\|C_n\|_F^2]=d^{n+1}$$ This ...
Yaroslav Bulatov's user avatar
1 vote
0 answers
53 views

Distribution of joint Gaussian conditional on their sum of squares

Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
Sheperd Lv's user avatar
7 votes
2 answers
335 views

Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation

$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
TheBestMagician's user avatar
2 votes
1 answer
120 views

Isolated eigenvalues of a random matrix

This is a continuation of this question. Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I want to study the eigenvalues of the matrix $O+O^\top + \lambda uu^\top$ where ...
Pluviophile's user avatar
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4 votes
1 answer
229 views

Spectral density of symmetrized Haar matrix

Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
Pluviophile's user avatar
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1 vote
0 answers
57 views

Limiting value of expectation of trace of truncated Gram matrix

Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
dohmatob's user avatar
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1 vote
0 answers
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Controlling quantity related to Laplacian pseudo-inverse of Erdős–Rényi graph

Consider an $n$-node undirected graph $G = (V, E)$ equipped with weights $W$. Let $L$ be the weighted graph Laplacian matrix, i.e. $L_{ij} = -W_{(i,j)}$ for $(i,j)\in E$ and $L_{ii} = \sum_{j:(i,j)\in ...
yy98's user avatar
  • 11
8 votes
1 answer
309 views

Why does this combinatorial sum vanish?

We define the coefficients $c_{k,k-i}$ of ${n \choose k}$ by the following easy expansion: \begin{align*} & {n \choose k} = \frac{1}{k!} n(n-1) \dots (n-k+1) = \frac{1}{k!} \prod\limits_{t=...
Tardis's user avatar
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