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Questions tagged [random-matrices]

For questions concerning random matrices.

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Distribution of Singular Values of Subunitary Matrix

Let $U$ be a random $n \times n$ unitary matrix (w.r.t. the Haar measure) and let $M$ be a $k \times l$ submatrix. What is the distribution of the singular values of $M$?
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Product distribution between random matrix and random vector

Let $D,d$ be integers greater or equal to $1$. Suppose that $A$ is a random-vector in $\mathbb{R}^{D\times d}$ (viewed as a random $D\times d$ dimensional matrix), and $X$ is a random vector in $\...
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Exercise 4.2.5 [HDP Book Vershynin]: Packing the balls into K

Suppose $T$ is a normed space. Prove that packing number $P(K,d,\epsilon)$ of $K \subset T$ is the largest number of closed disjoint balls with centers in $K$ and radii $\epsilon/2$. Show by example ...
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What does this physics paper mean by having a matrix in a denominator?

I am not familiar with any notation used in physics. The paper "Non-hermitian random matrix theory: Method of hermitian reduction" by Joshua Feinberg and A. Zee (Nuclear Physics B, 1997) states: A ...
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26 views

Expected value for the eigenvalue of random orthogonal matrix

I am interested in finding the expected value of the product of two eigenvalue with respected to the GOE ensemble. So I have a random $n\times n$ matrix with all off diagonal elements $\mathcal{N}(0,...
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Non-Negative irreducible matrices with random (correlated or independent) non-zero entries and deterministic zeros

Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $\lambda$, is positive and equal to its spectral radius $\rho(M)$. My question is ...
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Fix $v$, the probability that $v$ is orthogonal to any proper subspace of a random matrix $A$ with standard normal entries is $0$

Suppose $v \in \mathbb R^n$ is some fixed vector and $A \in M_n(\mathbb R)$ is a random matrix that each entry is generated by a standard normal distribution. I read a statement claiming: the ...
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Fix $v \in \mathbb R^n$, the probability matrix $A$ with normally distributed entries has an eigenbasis that $v$ projects nontrivially is $1$

Suppose $v \in \mathbb R^n$ is a fixed nonzero vector and $A \in M_n(\mathbb R)$ is a random matrix where each entry is taken from a standard normal distribution over $\mathbb R$. We know that the ...
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1answer
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Are these random matrix processes equivalent?

Suppose I would like to randomly fill an empty matrix (consisting of zeroes) with a particular number of each of the elements $a$ and $b$. One way is first fill the matrix with the proper number of $a$...
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Comparing Biased Random Walk models

Given a single graph, and two different "biased" random walk models on the same undirected graph, how does on theoretically compare the two models? What are the metrics one should theoretically study ...
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Total Variation Distance Between a Distribution and a perturbed distribution?

I'm interested in finding the total variation distance between the Gaussian Unitary Ensemble (GUE) over $n\times n$ Hermitian matricies and a perturbed version of it. Let the GUE have the measure $$ ...
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How to do change of variables of a j.p.d.f with N pdf(s)?

Given that I have a joint probability distribution(jpdf) of: $$P(x_1,...,x_N) = C_N \prod_{j=1}^{N}(1-x_j)^a(1+x_j)^b \prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ where $$\prod_{1\leq j <k \leq N} |...
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Expected value of eigenvalues of perturbed matrix

Let $A$ be an arbitrary $n \times n$ real matrix and $P$ be a random perturbation matrix with zero-mean, i.i.d. entries. Can we say anything about the eigenvalues of $A+P$? In particular, are there ...
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A statistical robust least-squares problem

Consider the following statistical robust least-squares problem $$\text{minimize } \mathbb E \|Ax - b\|_2^2$$ where $A = \bar A + U$, where $\bar A$ is the mean of $A$ and $U$ is a zero-mean ...
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1answer
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Approximate monotonicity of $\epsilon$-covering number

This is from Exercise 4.2.10 in Roman Vershynin's book, High-Dimensional Probability: An Introduction with Applications in Data Science. Let $(T;d)$ be a metric space and $N(A,d,\epsilon)$ be the $\...
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Upper bound on expected norm of subgaussian random matrix

Let $A \in \cal{M}_{n \times m}(\Bbb{R})$ be a random matrix with IID subgaussian entries with variance proxy $\sigma^2$. Show that $E[||A||_{op}] \le c \sigma \sqrt{m+n}$ for a constant $c$ to be ...
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Generating random regular matrices with some prescribed zeros

I am given a set $N\subset\{1,\dotsc,n\}^2$. I need to generate random invertible $n\times n$-matrices $A=(a_{ij})$ such that $a_{ij}=0$ if $(i,j)\in N$. The set $N$ is chosen such that this is always ...
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Studying Terence Tao's book on Random Matrix theory

Is the second chapter of Terence Tao's book on Random Matrix a good place to learn the basics of Random Matrix theory? I'm intrigued and have a long term view of trying to understand Tao & Vu's ...
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What is the product of two Haar distributed unitary matrices?

I guess a product of two Haar distributed unitary matrices is also a Haar distributed unitary matrix. Is there a proof?
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What is the expectation of the rank of a matrix with a 1 at each column?

Say a random square matrix $A\in\mathbb{R}^{n\times n}$, each column of $A$ has exactly one nonzero element being 1, i.e. each column looks like $e_i=\{0,\dots,1,\dots,0\}^\top$. Say for each column, ...
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Do imaginary inverses of non-invertible matrices exist?

There isn't a real solution to $x^2 = -1$, but a complex solution $x = i$ exists. Similarly, does there exist a complex inverse of non-invertible matrices?
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Bounding sub-Gaussian tail events by Gaussian tail events?

Background I have come across a problem where I want to derive a concentration inequality for sub-gaussian random variables. More precisely, I want to bound the spectrum of a certain empirical ...
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Concentration of the measure of a general covariance-like matrix

I consider a random matrix of the type : $M_n = \frac{1}{n} X_n D_n X_n^\intercal \in \mathbb{R}^{n \times n}$, in which all matrices are square of size $n$. $D_n$ is a deterministic diagonal matrix ...
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Question on random matrices

I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, \dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$. ...
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The rank of a matrix of Gaussian random vectors

If I generate a matrix $R\in\mathbb{R^{k\times p}}$, $k<p$, with i.i.d. entries $R_{i,j}\sim\mathcal{N}(0,\sigma^2)$, is there a guarantee that this matrix will have rank $k$, i.e. its columns will ...
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Reference request: Convergence of singular values of tall random matrix

Let $X_n\in\mathbb{R}^{n\times m}$ be a matrix whose entries are i.i.d. random variables with zero mean and variance $\sigma^2$. Let $m$ be a fixed integer and $\|\cdot\|$ denote the 2-norm of a ...
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Distribution of $(\langle X_i,X_j\rangle)_{i,j=1}^n$ for $X_k\sim\operatorname{Unif}(S^d)$

For two independent random variables distributed uniformly on a $d$-sphere surface ($X_1,X_2\sim\operatorname{Unif}(S^d)$), it is obvious that $$\langle X_1,X_2\rangle\sim -\langle X_1,X_2\rangle$$ by ...
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Expectation of $A^TXA$ for random $A$ and $X$

Suppose there are two random matrices (distribution unknown), denoted as $A$ and $X$, both in the $\mathbb{R}^{n \times n}$ space. It is known that $\|A\| \leq 1$ (for any $p$-norm) and $E[X]\geq0$. I ...
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largest singular value of gaussian random matrix

Let $A = A_{ij}, 1\le i\le n,1\le j\le m,$ be a random matrix such that its entries are iid sub-Gaussian random variables with variance proxy $\sigma^2$. Show that there exits a constant $C>0$ such ...
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Spectral measure of GOE matrices

I am studying a paper and there is a paragraph which describes some notation on random matrices.. I quote : " The GOE ensemble is a probability measure on the space of real symmetric $N\times N$...
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Total Variation Distance Between Map of Random Matrix and Original

Write $\Gamma(\boldsymbol{A}) = \boldsymbol{A} + \alpha\operatorname{diag}(\boldsymbol{A})$. Suppose $\boldsymbol{B}$ is a random matrix. Is it true that $$d_\mathsf{TV}(\boldsymbol{B}, \Gamma(\...
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Variance of norm of random matrix and sparse vector product

Let $M \in \mathbb{R}^{n*n}$ be a unitary matrix such that it's values are bounded by C. Let $x\in \mathbb{R}^{n}$ be an s-sparse vector. Let $R\in \mathbb{R}^{k*n}$ be a matrix with rows randomly ...
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Localization of eigenvalues

I have a naive question because it’s mentioned in every random matrix paper and is not explained. What does it mean to say a random matrix has localized eigenvalues? And what are some examples of it?
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Random Matrix Ensembles with no Quadratic Term

A large class of interesting physical problems (for example, 2D quantum gravity, matrix glass models etc.) are related to random matrix integrals with potential energy functions that have a leading ...
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Is there a distribution for random matrices which are constrained to have “unit vector” columns?

Let $a_{ij}$ be the components of an $n \times m$ random matrix of real numbers subject to the constraint that for each column $j$ we have: $$\sum_i{(a_{ij})^2}=1$$ In other words, consider a matrix ...
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Expectation in spectral sparsification algorithms

I am new to random matrices. I am studying the (Sampling) sparsification algorithms done by Daniel Spielman, Teng, Srivastava. They used the concept of graph sampling to obtain a good spectral ...
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Eigenvalues of random matrix

I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows: Let $A$ be random matrix s.t. $A=(...
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is random gaussian matrix invertible?

Is Gaussian Random Matrix invertible? I mean can we invert a Random Gaussian Square Matrix and also what is nature of its determinant, I mean to say whether determinant is zero or non zero?
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p-values from a wishart distribution

I am not a statistician but try to apply some multivariate statistics in order to find outliers (abnormal distances) in a distance matrix. Here is my problem: I have a $p \times p$ matrix $D$ of ...
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Identity between resolvent and singular value density

I was reading the paper Sengupta, Anirvan M., and Partha P. Mitra. "Distributions of singular values for some random matrices." Physical Review E 60.3 (1999): 3389. but got stuck at equation (3):...
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Trace inequality for a simple random matrix

I have two random vectors $x,y \in \mathbb{R}^n$, with joint probability law $P(x,y)$, and each has zero mean, $\mathbb{E} x = \mathbb{E} y = 0$. I build the rank-one matrix $M = xy^\intercal$. My ...
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Farkas lemma and matrix spectrum

I am currently looking at a problem of the following type : I have a matrix $\mathbf{M}\in\mathbb{R}^{N \times N}$ such that it's general term is given by $(\mathbf{M})_{ij}=z_i \delta_{ij} - A_{ij}...
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Probability of a quantity from a nasty jpdf

I'm doing a project on random matrices and its applications. I have the joint probability density of Ginibre ensemble and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have $...
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Are the eigenvectors of real Wigner matrices made of independent random variables with zero-mean?

I am trying to understand a portion of this paper [p. 3] and got stuck in the following statement, which sounded kind of trivial, but has been deceiving me for a while. Would you help me understand? ...
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Eigenvalues of traceless random matrix

For random matrices we know the eigenvalues have the Wigner semicircle distribution. I wonder what is the distribution of eigenvalues of traceless random real symmetric or Hermitian matrices? This is ...
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Injecting cross-sectional and temporal correlations into a random matrix

Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,\ldots,4$, $t=1,\ldots,3$. $$X=\left( \begin{array}{cccc} x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \\ x_{1,...
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Image of Gaussian random vector under linear map

I would like to prove the following lemma: Given a $n\times n$ matrix $A$, gaussian random vector $x$, and constant $\mu\in(0,1)$, $$ \mathbb{P}( \Vert Ax \Vert / \Vert x \Vert < \mu \Vert A \...
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Why are independence and mean-zero necessary for the symmetrization lemma to hold?

I'm going through the proof of the symmetrization lemma (Vershynin), which says $$ \frac{1}{2} E \left\| \sum_i \epsilon_i X_i \right\| \leq E \left\| \sum_i X_i \right\| \leq 2 E \left\| \sum_i \...
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Determinant of a random symmetric matrix

I am trying to determine the determinant of the following uniformly distributed random symmetric matrix $A$ with zero mean. \begin{equation} A= \begin{pmatrix} 1 & \cos \alpha_{12} & \cos \...
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How can the Wigner semicircle distribution go to zero?

The Wigner semicircle distribution takes a form that goes to zero beyond a certain R. I don't see why it can actually hit zero however. According to Wikipedia's article on the Wigner semicircle ...