# Questions tagged [random-matrix]

A random matrix is a matrix whose entries consist of random variables from some specified distribution. Random matrices have many modern applications in physics, finance, statistics and numerical analysis.

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### Stable Law for products of random matrices?

Product of random variables tends to converge to Log-Normal distribution, is there a similar law for a product of large invariant random matrices? Following Burda survey, Section V, multiplying such ...
17 views

### Using Dudley Integral to estimate maximum singular value of Gaussian random matrices [duplicate]

On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality. Can we use ...
59 views

### Covariance of a vectorized random matrix

I am looking for an answer to the following question. Assume a random matrix $\mathbf{A}$ of dimension $n\times n$ such that each row $\mathbf{A}_i$ of the matrix $\mathbf{A}$ is a realization of a ...
158 views

### Inverse of Matrix with normally distributed Elements?

Let's say we have some Matrix $X \in \mathbb{R}^{n_x \times n_x}$ whose elemets $x_{i,j} \sim \mathcal{N}(\mu, \sigma^2)$ are normally distributed. In other words: $vec(X) \sim \mathcal{N}(m, S)$, ...
81 views

### Questions on the Wishart distribution

If $X$ is an $n\times p$ matrix where each row is iid multivariate normal, then $X^TX$ has a Wishart distribution. What is known about the limiting distribution of $X^TX$ for large $n$ when the rows ...
1 vote
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### Random matrix theory and research - a lot like doing linear algebra?

I've been searching for a subfield of research to get into and wonder whether random matrix theory could suit me well; it seems like it does, because the stuff I read, and the seminars that I watch ...
31 views

### Minimax equivalence of Matrix Norm

In a proof of random matrix theory, the author makes use of the following equivalence: \begin{equation} \inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv \end{equation} ...
27 views

### Factorizing a matrix of distributions [closed]

Let's say we have a matrix $X \in \mathbb{R}^{m \times n}$, then the (R-truncated) SVD allows to approximate: $X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j}$ Now I ...
1 vote
595 views

### marchenko pastur for Correlation

It has been suggested to me that if I construct a covariance or correlation matrix using factor model then I can use the Marchenko-Pastur distribution to highlight significant correlations (or ...
819 views

### How to estimate the largest eigenvalue of a correlation matrix from one observation of underlying data matrix?

Suppose that I have $N$ time series $x_{1t},x_{2t},\dots,x_{Nt},$, that are correlated with each other. A $N\times N$ correlation matrix is $R=\rho_{ij}$. It can be represented with eigen value ...
161 views

### Marginal Distribution of Matrix Normal with Two Inverse Wisharts

Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions $$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$\Sigma \sim IW(a, A)$$ $$\Omega \sim IW(b, B)$$ where $a$ and $b$ are ...
166 views

### Are there random matrix results like Marcenko-Pastur, but for CCA?

The Marcenko-Pastur law is about asymptotic distributions of eigenvalues. It starts from a simple null model (iid zero-mean Gaussian entries) and derives a distribution for the spectrum. In PCA, this ...
133 views

### How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the eigenvalue distribution using whatever you can. Background In my field, I have a ...
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### Expectation of a Matrix Raised to a Power

I have been able to find little literature related to the topic online (e.g., A note on the Expected Value of an Inverse Matrix such that E($\frac{1}{X})$ $\geq$ ($\frac{1}{E(X)}$), though, vaguely, I ...
1 vote
200 views

### Rank of N x D vs D x N matrices

If $X$ is a random $N \times D$ matrix where $N > D$, then why is the rank of X - mean(X, 1) $D$ while the rank of ...
94 views

### Distribution of $\mathbf{A}\mathbf{X}$?

Let $\mathbf{A}$ be an $m\times n$ random matrix with entries $A_{ij}$ being jointly Gaussian. Suppose all of these variables are independent of the random vector $\mathbf{X} = (X_1,\ldots,X_n)^\top$ ...
181 views

### conjugate prior (and posterior) for Matrix Variate Distributions

I was wondering what is the conjugate prior for Matrix Variate Distributions (e.g., unknown mean, known variance matrices), and what's the corresponding posterior? Is there analytical solution?
475 views

### Distribution of eigenvalues of a random matrix

Given a random symmetric matrix A whose entries are Poisson distributed, can anything be said about the distribution of A's eigenvalues? Would be great if someone could link a paper citing such a ...
3k views

### If I generate a random symmetric matrix, what's the chance it is positive definite?

I got a strange question when I was experimenting some convex optimizations. The question is: Suppose I randomly (say standard normal distribution) generate a $N \times N$ symmetric matrix, (for ...
1 vote