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1 vote
0 answers
74 views

Asymptotically small submatrices of random matrices

Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure $$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$ for ...
8 votes
1 answer
428 views

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 $D$ ...
1 vote
0 answers
43 views

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, ...
9 votes
2 answers
496 views

Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?

For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...
5 votes
1 answer
401 views

Lower tail of random rank one sums?

Let $\{x_i\}_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$; assume that $\|x_i\|_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x_1 \otimes x_1]$. Define $$ \Sigma_n =...
2 votes
0 answers
269 views

Singular values of Kronecker product of random matrices

I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
0 votes
0 answers
112 views

Additivity of purity of random matrix products

Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as $$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
1 vote
0 answers
68 views

Limiting value of expectation of $\operatorname{tr}(BR(z))$, where $R(z) := (X^\top X - z I_d)^{-1}$ and $X \sim N_{n,d}(0,A)$

Let $A=A(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X$ be an $n \times d$ random matrix with iid rows from $N(0,A)$. Let $R$ be the resolvent of ...
5 votes
1 answer
282 views

What is the spectral norm of a random projection times a diagonal?

Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix. What can be said about the distribution of the largest ...
1 vote
1 answer
91 views

Density of eigenvalues of empirical covariance matrix of vectors uniform on the sphere

Is anyone able to point me to a reference for this? Let the rows of $X \in \Re^{n\times d}$ be i.i.d. uniform on the sphere of radius $\sqrt{d}$ in $\Re^d$. What is the density of the eigenvalues of $...
2 votes
2 answers
206 views

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent). Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...
10 votes
3 answers
2k views

Random walks and Lyapunov exponents

Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $\mathrm{GL}_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y_1\|)$ is finite, there exists a ...
15 votes
1 answer
1k views

Has the technique of "sprinkling" been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
10 votes
1 answer
492 views

(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$

Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits_{i=0}^{\infty} A^{2i}=\left( ...
1 vote
1 answer
160 views

Estimates of product of eigenvalues gaps for Wigner matrices

Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$. My question. For any fixed $i\in\{1,\dots,n\}$, ...
0 votes
0 answers
133 views

Spectral CLT for random matrices with iid entries

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\...
1 vote
1 answer
286 views

Bound on $i$th largest eigenvalue in a large Erdos-Renyi graphs

Typical magnitude of $i$th largest eigenvalue of an Erdos-Renyi random graph seems to decay at least exponentially with $i$. Is there an analytic expression for the constant in the exponent, or a nice ...
0 votes
1 answer
209 views

Factorisation of Gaussian random matrix into random Hermitian and correction factor

By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries $$\mathbf{\Gamma}_{n\times k}...
27 votes
3 answers
13k views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
2 votes
1 answer
212 views

Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$

Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
0 votes
1 answer
280 views

Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
0 votes
0 answers
141 views

What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$

Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
2 votes
1 answer
1k views

Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data

Let $Z=[z_1, \dots z_n]$ be a $d \times n$ matrix, where the $z_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $...
1 vote
0 answers
124 views

Law of large numbers and Central Limit Theorem for eigenvalues of perturbed matrices

I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely, Let $ \forall n \in \mathbb{N},...
1 vote
0 answers
83 views

Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?

Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
0 votes
0 answers
115 views

Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?

Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
1 vote
1 answer
104 views

Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?

Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...
3 votes
1 answer
3k views

Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
13 votes
4 answers
1k views

Why only three classical matrix ensembles in random matrix theory?

I am just starting out on understanding random matrix theory from a background in applied mathematics. I have a very basic question about the Gaussian ensembles: why are there only three classical ...
4 votes
2 answers
2k views

Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form $$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix} $$ where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
4 votes
1 answer
372 views

Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
1 vote
0 answers
175 views

Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
32 votes
3 answers
12k views

What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
1 vote
0 answers
143 views

A "conjectured" concentration inequality for operators, probably related with random matrix theory

I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as ...
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 ...
4 votes
2 answers
2k views

Advanced reference and roadmap about random matrices theory

There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question. I really want to hear ...
3 votes
0 answers
98 views

Asymptotic results on statistical graph models

This post is partly inspired by this post. Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix While it is well-known that two basic ...
2 votes
1 answer
2k views

Bounds on the eigenvalues of the covariance matrix of a sub-Gaussian vector

Suppose that $\boldsymbol{x}\in\mathbb{R}^n$ is subgaussian random vector of variance proxy $\sigma^2$, i.e., $$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\...
10 votes
4 answers
645 views

Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
14 votes
1 answer
449 views

References for reasoning about the spectrum of a convex body?

By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
2 votes
0 answers
54 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
10 votes
1 answer
1k views

Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
0 votes
1 answer
184 views

Norm of matrix with randomly deleted entries

Let $A$ be an $n \times n$ matrix with real entries and let $B$ be the random matrix whose $(i,j)$ entry is $$B_{i,j}=v_{i,j}A_{i,j}$$ where the $v_{i,j}$ are i.i.d Bernoulli random variables with $P(...
6 votes
0 answers
352 views

How to generate a random (Weyl) curvature operator ?

Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity : ...
5 votes
1 answer
312 views

Expected inverse determinant with independent rows

Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors. More ...
9 votes
2 answers
4k views

Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix

TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...