# 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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### Singular values of $A^{-1}$ where $A$ has standard normal entries

Is anything known about behavior of singular values of $A^{-1}$ where $A$ has standard normal entries? Empirically, $k$th singular value appears to decay as $1/k$, is this well-known? colab
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### What is the difference between a matrix normal distribution and the multivariate gaussian distribution?

$\newcommand{\vec}{\operatorname{vec}}$Consider a set of $N$ matrices $X_1, X_2, \ldots, X_N$. I want to estimate the distribution of these matrices represented by the mean and covariance. I address ...
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### On distributions over orthonormal sets: existing families, construction, and simulation

Have families of distributions over orthonormal sets been defined and studied in the literature? What are a couple examples and/or references? Are there known methods for constructing distributions ...
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### Distribution of a product of a matrix with a random matrix

Suppose we have the matrices $Z\in \mathbb{R}^{n\times n}$ and $X\in \mathbb{R}^{n\times d}$, such that each row $x_i\in\mathbb{R}^d$ is drawn i.i.d from a $N(0,\Sigma_{d\times d})$ distribution. ...
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Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define $$\mathbf{S} = \... • 21.3k 2 votes 1 answer 202 views ### Random matrix theory and research - a lot like doing linear algebra? I've been searching for a subfield of research to get into and wonder whether random matrix theory could suit me well; it seems like it does, because the stuff I read, and the seminars that I watch ... 0 votes 0 answers 34 views ### Minimax equivalence of Matrix Norm In a proof of random matrix theory, the author makes use of the following equivalence: $$\inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv$$ ... 2 votes 0 answers 28 views ### Factorizing a matrix of distributions [closed] Let's say we have a matrix X \in \mathbb{R}^{m \times n}, then the (R-truncated) SVD allows to approximate: X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j} Now I ... 1 vote 1 answer 804 views ### marchenko pastur for Correlation It has been suggested to me that if I construct a covariance or correlation matrix using factor model then I can use the Marchenko-Pastur distribution to highlight significant correlations (or ... • 1,317 3 votes 1 answer 961 views ### How to estimate the largest eigenvalue of a correlation matrix from one observation of underlying data matrix? Suppose that I have N time series x_{1t},x_{2t},\dots,x_{Nt},, that are correlated with each other. A N\times N correlation matrix is R=\rho_{ij}. It can be represented with eigen value ... • 61.8k 4 votes 0 answers 190 views ### Marginal Distribution of Matrix Normal with Two Inverse Wisharts Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions$$X \sim MN_{p\times n}(0, \Sigma, \Omega) \Sigma \sim IW(a, A)  \Omega \sim IW(b, B)$$where a and b are ... • 978 3 votes 1 answer 198 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 ... • 5,219 3 votes 1 answer 146 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 ... 5 votes 1 answer 604 views ### Transformation of Inverse Wishart Let \Sigma be an p\times p dimensional covariance matrix that is distributed Inverse Wishart with degrees of freedom \nu and Prior scale matrix \Psi such that we write \Sigma \sim W^{-1}(\nu, ... • 978 8 votes 2 answers 174 views ### Random rotation of a set of distinct points in R^n Consider a set \{\mathbf{X}_1,\cdots , \mathbf{X}_M\} of distinct points in \mathbb{R}^n with M finite. The M values of the i-th coordinate do not all have to be dinstinct. For example, in \... • 667 2 votes 0 answers 230 views ### 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 ... • 161 1 vote 1 answer 237 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 ... • 3,233 4 votes 0 answers 95 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 ... • 61 2 votes 0 answers 209 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? • 241 3 votes 0 answers 488 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 ... • 157 33 votes 1 answer 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 ... • 37.2k 1 vote 2 answers 3k views ### Variance of Random Matrix Let's consider independent random vectors \hat{\boldsymbol\theta}_i, i = 1, \dots, m, which are all unbiased for \boldsymbol\theta and that$$\mathbb{E}\left[\left(\hat{\boldsymbol\theta}_i - ...
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Does anyone have a reference or proof for the LTE and LTV for matrices? I am defining the unconditional variance for matrices in the usual way:  \operatorname{Var}_{m}(M) \overset{\text{def}}{=} \...