Questions tagged [random-matrices]

For questions concerning random matrices.

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35 views

How to estimate this (unusual?) model for a time series of covariance matrices?

I am wondering if anyone can point me to econometrics or statistics literature on estimating the covariance matrices of models of the following form. Or, if someone with enough domain knowledge would ...
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1answer
36 views

Is the inequality of the random matrices correct?

I am not familiar with random matrices but I need to confirm the correctness of the inequality below. Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let $A_1,\ldots, A_n$ be $m\times m$ ...
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23 views

How extreme singular values affects least square for Gaussian covariates.

Consider the generative model for linear regression w.r.t. the true parameter $w^* \in S^{d-1}$ $$y=Xw^*+e$$ with i.i.d. Gaussian error $e \sim N(0, \sigma^2I_n)$. Let $X \in \mathbb{R}^{n\times d}$ ...
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137 views

A characterisation of Wishart distribution when rows of $\textbf{X}(n\times p)$ are i.i.d. $\mathcal{N}_p (\mu\textbf{1}, \Sigma)$

If the rows of $\textbf{X}(n\times p)$ are i.i.d. $\mathcal{N}_p (\mu\textbf{1}, \Sigma)$ and $\textbf{C}$ is symmetric matrix then $\textbf{X}'\textbf{C}\textbf{X}$ has a $\mathcal{W}_p(\Sigma, r)$(...
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16 views

Why is the Jacobian of this random vector transformation the determinant of the fixed matrix?

Here's a link: https://www.probabilitycourse.com/chapter6/6_1_5_random_vectors.php. My question concerns example 6.15 Here is the stated problem: Let $\mathbf{X}$ be an $n$-dimensional random vector. ...
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28 views

Matrix Inverse and $O_p$ Notation

I made an attempt at proving the following proposition but am not sure if it is correct. Any feedback would be greatly appreciated. Proposition. Let $(X_n)$ be a sequence of random $k\times k$ ...
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11 views

Differentiation wrt eigenvalue: interchange integration and differentiation; chain rule

This question are actually two (related) questions. Let $f$ be a scalar valued function that depends on some random variable $X$ and smooth on a deterministic $n\times n$ symmetric matrix $A$ with $\...
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1answer
19 views

Wigners semi circle law from the Stieltjes transform

I struggle to complete the last step of the derivation of Wigner's semi circle law (or the Marcenko-Pastur density for that matter), from the corresponding Stieltjes transform. The Stieltjes ...
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17 views

Real and Symplectic versions of a special case of the Itzykson-Zuber integral

Itzykson-Zuber integral is given by $$ I = \int_{U(N)} d U \mathrm{e}^{\operatorname{tr}(A U B U^\dagger)} = \left(\prod_{p = 1}^{N-1} p! \right) \frac{\mathrm{det}\left(\mathrm{e}^{a_i b_j}\right)}{\...
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25 views

Spectral norm of random matrices times a diagonal matrix

Consider some scalars $d_1\leq \dots\leq d_k$ and let $D = \text{diag}(d_i)\in\mathbb{R}^{k\times k}$. Assume that $X\in \mathbb{R}^{k \times k}$ is a random matrix of i.i.d. normal entries (i.e. $X_{...
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31 views

Projection of a random i.i.d matrix

Let $A$ be a random matrix with i.i.d entries. Under what conditions does $CA$ has i.i.d entries? Particular interest is (a) when $C$ is fixed deterministic matrix and (b) When $C$ is i.i.d random. ...
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12 views

how to do Stieltjes transform of this example

Stieltjes transform for a measure $u(x)$ is given by, $$S(u,z)=\int \frac{d u(x)}{z-x} $$ If $supp(u)=R$ then the Stieltjes transform is analytic in both $/{z\in C:lm z>0/}$ and $/{z\in C:lm z&...
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Family of sparse ill-conditioned matrices

I am wondering if there is a well-known natural family of sparse ill-conditioned matrices. Specifically, I am looking for a family of (random) matrices $A \in \mathbb{R}^{n \times d}$ where $\lambda_{\...
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14 views

Smallest singular value of the sum of i.i.d. random rectangular matrices

Suppose $E_i$'s are iid random matrices (not square) such that $\mathbb{E}[E_i] = 0$ and $\mathbb{E}[E_i^TE_j] = 0$ for $i\ne j$. Let $M = \sum_{i=1}^n E_i$. It is clear that $$ H^* = \mathbb{E}[M^...
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Deterministic matrix with prescribed limit distribution

I have a question about a common argument in free probability theory. Often times, when we want to prove a statement about free-ness or other properties of distributions of elements in a non-...
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22 views

Probability distribution of eigenvalues of a random matrix

I was trying to numerically find the probibility distribution of the eigenvalues of a matrix whose elements are random within the range [-0.5:0.5]. In every case, I found that the distribution looks ...
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49 views

Solve $A\begin{bmatrix}1&0\\0&0\end{bmatrix}=\begin{bmatrix}1&0\\0&0\end{bmatrix}A,\qquad\left(A+\begin{bmatrix}0&1\\1&0\end{bmatrix}\right)^2=I_2$ [closed]

Solve the matrix equations $$A\begin{bmatrix}1&0\\0&0\end{bmatrix}=\begin{bmatrix}1&0\\0&0\end{bmatrix}A,\qquad\left(A+\begin{bmatrix}0&1\\1&0\end{bmatrix}\right)^2=I_2,$$ that ...
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Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables

Having no answer so far, I asked this on MO now. My question is motivated by this question and this question, where the first was aimed for giving a one sided tail bound for maximum of subgaussians, ...
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12 views

Distribution of eigenvalues under product with Gaussian matrix

Let $0\le \alpha_1\le \dots \le \alpha_N$ be the eigenvalues of matrix $A^\top A$, where $A\in R^{N\times N}$ is an arbitrary matrix. Define the product $\textbf{P} := A \textbf{W}$, where $\textbf{W}\...
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1answer
34 views

Rank-1 random matrix is positive definite?

Let $X\in\mathbb{R}^n$ be a random vector with continuous coordinates such that $\|X\|_2=1$ a.s. Define the random matrix $A=XX^T$. It is obvious that $A$ has rank 1 and its unique non-zero eigenvalue ...
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1answer
19 views

Gaussian orthogonal ensemble and Haar measure

I have been struggling with a probably easy question but I cannot prove it, so any insights would be really helpful. If I have a random matrix $X \in GOE(N)$, namely from the Gaussian orthogonal ...
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1answer
81 views

norm of i.i.d random matrix with small perturbation.

Let $A=(a_{ij})_{ij}$ be an $m \times n$ random matrix with i.i.d entries. Let $E=(\epsilon_{ij})$ be a deterministic $m \times n$ matrix with $|\epsilon_{ij}|<\epsilon$ where $\epsilon$ is a fixed ...
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18 views

Genes mirror geography on a torus?

Disclaimer: this is an open-ended, imprecise question, asking for speculation in a topic that I know relatively little about (random matrix theory and principal component analysis). I wanted to air ...
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1answer
66 views

Concentration of norm of linearly transformed normal random vector as dimension go to infinity

Following no response, recently asked on MO. Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $Bx_k:= x_k/ {k}, 1 \...
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25 views

MAPLE: How do I make this generation of a list of random matrices NOT include the word generate in front of each number?

my code in MAPLE says H:= Generate(list(RandomMatrix(7,3),4)); which does generate four random matrices but before each number in each matrix it says "Generate". ...
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1answer
32 views

Eigenvalues of an iterative and random selection between two different linear transformations.

I was wondering... Consider a couple of different square matrices that act as linear transformations. I am going to apply these two linear transformations in an iterative manner, but I will select one ...
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25 views

Given $A \in \mathbb R^{m \times n}$, find upper bound for $\mathbb E\|Az\|_q$ for $z$ drawn uniformly at random on the sphere $\{\|z\|_p = 1\}$

Let $m$ and $n$ be positive integers and $p,q \in [1,\infty]$. Consider the finite-dimensiaonal normed vector spaces $X = (\mathbb R^m,\|\cdot\|_p)$ and $Y = (\mathbb R^n,\|\cdot\|_q)$, where $$ \|x\|...
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Question on covariance estimation (matrix concentration inequality)

This is a question arising from Vershynin's high-dimensional probability book. In that book, Theorem 4.7.1 (Covariance Estimation Theorem) states that Let $X$ be a subgaussian random vector in $\...
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2answers
58 views

Definition of functions applied to a matrix

I am reading a paper (on random matrix theory) and its using a lot of notation like this: Given a Hermitian matrix $H$ and an approximate $\delta$ function $\theta_\eta(x) = \frac{\eta}{x^2 + \eta^2}...
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1answer
35 views

why is this conditional PDF of a random variable a delta function… or not?

I was reading page 11 of an introduction to random matrices; in particular, this piece: Why is (2.2) true? It seems reasonable, but I think this delta function is confusing me. I want to break the ...
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1answer
15 views

Gaussian width after some linear transformation

The Gaussian width $w(T)$ of a set $T\in \mathbb{R}^n$ is defined as follows: $$ w(T) = \mathbb{E}\sup_{x\in T} \langle g,x\rangle $$ where $g$ is a random normal vector in $\mathbb{R}^n$. The ...
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Can properties for (circular symmetric) complex random matrices automatically work for real random matrices?

I am dealing with a theorem which relates to circularly symmetric complex Gaussian random matrices (CSGRM). It seems quite tempting to assume that the theorem also extends to real-valued Gaussian ...
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Expectation of sum of last $k$ eigenvalues of $\mathbf{A}\mathbf{A}^T$

Supposing the eigenvalues of $\mathbf{A}\mathbf{A}^T$ are $\sigma_1,\sigma_2,\cdots,\sigma_{k+1},\cdots\sigma_{n+k}$($\sigma_1\leqslant\sigma_2\leqslant\cdots\leqslant\sigma_{k+1}\cdots\leqslant\...
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19 views

Two inequalities involving the resolvent of sample covariance matrix

Let $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$ be the data matrix where the $x_i \in \mathbb{R}^{p \times 1}$. Let the data be "centered at origin", so $\frac{1}{n}\sum_{i=1}^{n}x_i = 0$. Let $...
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25 views

Eigenvalues of random stochastic matrix

Let $G=(V,E)$ be some random graph with $|V|=n$, $|E|=m$ and let $\mu\in (0,1)$. Define the random stochastic matrix $P$ of size $n\times n$ for $i,j\in V$ via $P_{ij} = \begin{cases}1- \mathrm{deg}(...
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Is it a (minor) typo in the proof of Roman Vershynin's “High dimensional probability with application to data science” (linked), Theorem 3.1.1

So I've been currently studying this book on high dimensional probability by Roman Vershynin, which I find pretty awesome! However, I was wondering if in the first line of the proof of Theorem 3.1.1 (...
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Limiting empirical spectral distribution of random symmetric matrices with zero diagonal

Let $E_n \equiv E$ be an $n \times n$ random matrix so that: (1) $E_{ii} = 0 \forall i $ (2) $E_{ij}= E_{ji} \forall j \ne i$ (3) $E_{ij} \sim P$ (common probability law) are iid with mean $0,$ ...
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Multidimensional integral with the Laplace method

Define $ \mathbb{W} := (W_{i, j})_{1 \leq i, j \leq k} $ and \begin{align*}%$ \varphi_{C, k}(\mathbb{W}) := C^2 \, \boldsymbol{1}_k^T (I_k + \mathbb{W} )^{-1} \boldsymbol{1}_k + \sum_{1 \leq i < j \...
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Possible application of Laws of Large Numbers and Central Limit Theorem to SVD of dual covariance matrices

Let $X:=[x_1\dots x_n] \in \mathbb{R}^{d\times n}, x_i \in \mathbb{R}^{d\times 1}$ be a data matrix where $x_i \in \mathbb{R}^{d\times 1}$ are iid random vectors with mean $\mu$ and covariance $\...
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Why am I experimentally getting more small eigenvalues of sample covariance matrices when data come from lower dimensional spaces, and contrary?

I'm new to Random Matrix Theory (RMT) and I'm playing around (=coding) with generating data in low dimensions, nonlinearly embedding them into a high dimension, and I'm noticing a seeming intuitively ...
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2answers
25 views

Parametrization of a matrix drawn randomly from $SU(n)$ (using Haar measure)

I have been trying to find a (simple) parametrization of a random Unitary matrix, drawn from $SU(n)$, in terms of random variables. A trivial example would be a matrix drawn from $U(1)$, $$M = [e^{i\...
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1answer
25 views

Neumann series with random matrices

Suppose that we have a sequence of random square matrices $\{A_n\}_{n\ge1}$ that converge to $0$ in probability as $n\to\infty$, i.e. $\|A_n\|\to0$ in probability as $n\to\infty$, where $\|\cdot\|$ is ...
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1answer
37 views

A result on power of random symmetric matrices and vector multiplication: u^T S^n v

I have a rough intuition about some results regarding random matrices theory, but I'm not sure if this is correct - if it exists already somewhere or if there is a way to prove it clearly. Given: $n,...
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18 views

Will the unit circle of the circular law of random matrices still hold if the entries are not scaled to mean 0

The circular law states that if a random matrix is sufficiently large enough and the entries of the matrix is scaled to have mean 0 and variance 1/n, then the distribution of the matrices' eigenvalues ...
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1answer
43 views

How to generate a random vector with fixed sum and bounded elements

How we can generate a random vector $E =[e_1, e_2,e_3.\dots, e_N] \in R^N$ such that $\sum_i^N e_i = T \;$ and $ 0 \le e_ i \le d_i$ $\forall i \in 1,2,3,\dots,N$ where $d_i$ specifies the ...
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23 views

Concentration for the maximum magnitude entry for a random matrix

For $A \in {\mathbb R}^{n \times m}$ let, $$ \vert A \vert _{\max} = \max_{\substack{i=1,\ldots, n\\ j = 1,\ldots,m}} \vert A_{ij} \vert $$ Are there any concentration inequalities known about this ...
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1answer
33 views

Smallest singular value of product of 2 random matrices

Let $A\in\mathbb{R}^{n\times m}$ and $B\in\mathbb{R}^{m\times k}$ be two random matrices (each element is drawn iid from a normal distribution). Also $n<m<k$. Let $\sigma_{min}(A)$ be the ...
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27 views

Concentration inequality on 1-norm of random vector

I would like to give an upper bound on $\Pr\{||X-\mathsf{E}[X]||_1 > t\}$ where $X$ is a $d$-dimensional random vector with each entry follows i.i.d. binomial $(n,p)$ (so $\mathsf{E}[X]$ is ...
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1answer
67 views

Expectation of the maximum eigenvalue.

Let $X_1, X_2, \ldots X_n$ identically distributed independent random variables with zero mean and finite variance $\sigma^2$. Let $\bar X$ be the sample mean and consider the random vector $ B= n^{-1/...
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42 views

How to define the expectation and covariance of a random variable taking values in an inner product space?

I'm dealing with random variables that takes values in an inner product space, that is possibly finite dimensional but not necessarily Euclidean. To be precise: let $dim(V)=d$, and we equip $V$ with ...

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