# 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, ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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\...

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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 $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...

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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 $D$ ...

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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 ...

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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 ...

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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^\...

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### 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^{...

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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 $\...

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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 =...

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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 ...

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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^\...

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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}, \...

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### 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 $...

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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 ...

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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 ...

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### 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 ...

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### 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 $R^n$, ...

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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 ...

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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 ...

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### 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} \...

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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 ...

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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 ...

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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 ...

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### 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 ${X^T X}$...

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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 ...

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### 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
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...

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### 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 ...

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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 ...

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### 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 ...

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### 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 $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
...

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### 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 ...

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### 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 ...

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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 $...

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### $\log\det$ 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 ...

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### 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?

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### 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 ...

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### 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 $\|\...

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### 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://...

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### 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 ...

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### 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'...

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### 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 $v \in \mathbb R^d$, ...

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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 ...

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### 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=...