# Questions tagged [random-matrices]

For questions concerning random matrices.

549 questions
Filter by
Sorted by
Tagged with
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 ...
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 $m\times m$ ...
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 $X \in \mathbb{R}^{n\times d}$ ...
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 $\mathcal{W}_p(\Sigma, r)$(...
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. ...
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 $k\times k$ ...
11 views

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 ...
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 ...
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 ...
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 ...
66 views

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 ...
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 ...
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 ...
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 ...
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 ...