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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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} ...
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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 ...
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14 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 ...
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29 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 ...
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23 views

Would randomly permuting a quasirandom sequence successfully avoid otherwise intrinsic correlations in estimation?

Let $A$ be an $n \times n$ (Ginibre) matrix of complex-valued entries, the real and imaginary parts of which are, thus, standard normal variates, and $U$ be an independent such matrix, the rows and ...
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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 ...
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18 views

On expected values of triple Kronecker Products

Consider a random vector $\boldsymbol{x} \in \mathbb{R}^N$ and the identity matrix $\boldsymbol{I}_N \in \mathbb{R}^{N\times N}$. I have to compute the expected value of the following Kronecker ...
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How are singular values related to minimum mean squared error?

When I read the Sergio Verdu's tutorial on Eigenvalues and Singular Values of Random Matrices: A Tutorial Introduction, I found a very interesting conclusion on Slide 5-7, which says that, for a ...
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1answer
39 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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60 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, ...
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51 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 $\...
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63 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 ...
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1answer
19 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 ...
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86 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?
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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 ...
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2k 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 ...
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2answers
551 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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300 views

Law of Total Expectation and Law of Total Variance for Matrices

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}}{=} \...
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35 views

Eigenvalue order of magnitude for Wishart random matrix

If we have a $P\times N$ matrix $\mathbf{A}$ whose elements $A_i$ are samples (in this case, P samples) from a multivariate gaussian distribution in an $N$ dimensional space, we can define the Wishart ...
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Why did statisticians define random matrices?

I studied mathematics a decade ago, so I have a math and stats background, but this question is killing me. This question is still a bit philosophical to me. Why did statisticians develop all sort of ...
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107 views

Product of two random Gaussian matrices - orthant probability

In my studies of random matrices, I recently came across this challenge: Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., ...
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Marginal distribution of distribution on the Unitary group

I am trying to understand the behavior of distributions over the Unitary group (i.e. the set of square matrices $P$ such that $P^tP = I_d$), or in general distribution over the Stiefel manifold (set ...
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1answer
172 views

Does this method uniformly sample 3x3 rotation matrices?

Let $\textbf{X}=(X_1,X_2,X_3)$ be a random unit vector in $\mathbb{R}^3$ such that $\textbf{X}$ is uniformly distributed on the unit sphere $S^2$. Next, let $\textbf{Y}=(Y_1,Y_2,Y_3)$ be a random unit ...
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Intuitive explanation for Marchenko-Pastur law

I am looking for an intuitive reasoning behind the Marchenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
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Hypothesis testing for matrices of RVs

I looking for the correct statistical test for comparing matrices of random variables - my IV being a categorical variable and my DVs being n by n matrices of random variables (which are themselves ...
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1answer
129 views

Write $\mathop {\mathbb E}[(X AX^h)]$ in function of $\mathop {\mathbb E}[(X X^h)]$?

Suppose $X$ is an $i \times j$ random matrix. In addition, $X$ has complex i.i.d. normal entries with $0$ as mean. We define $A$ (of dimension $j \times j$) as a deterministic matrix. Is it ...
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1answer
91 views

How to check if a distribution has undefined variance?

How can I determine if experimental data comes from a distribution where the variance is undefined (e.g. the Cauchy distribution)? I honestly have no idea how to attack this problem in a sensible way,...
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2answers
1k views

Convergence of eigenvectors and eigenvalues of matrix that converges

For each random variable $X=x$, there is a symmetric positive definite matrix $M(x)$. Suppose there is a set of samples of random matrix $M_1,M_2,...,M_n$, where each $M_i$ is calculated based on the ...
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149 views

Parametric distributions over orthogonal projection matrices?

Consider the set of rank $m$ orthogonal projection matrices in $\mathbb{R}^{d\times d}$, for some $m<d$. With appropriate measure-theoretic considerations, one can define multiple distributions on ...
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1answer
165 views

Smoothing Kernel Preventing Simulation of Semi-Circle Law of Random Matrices

In trying to understand the properties of random matrices in the book "Plane Answers to Complex Questions" by R.Christensen, I came across the Semi-Circle Law, and tried to reproduce it with ...
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689 views

Eigenvectors of a Wishart matrix

I have been trying to find a good source (or clarifications) to help me understand this point. I am very new to random matrix theory so any pointers will be appreciated. Here is what I think I have ...
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1answer
104 views

Spectral norm of a sparse Gaussian matrix

Suppose $G$ is an $m \times n$ matrix such that each entry of $G$ is a standard normal variable. We know that the spectral norm of $G$ scales as $\sqrt m + \sqrt n$. Now, given a set of indices $S$ ...
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241 views

Difference between recentred and scaled eigenvalues and the Tracy Widom distribution

I have been generating correlation matrices from independent normal data simulated using the MASS package. I do this k times and extract the eigenvalues of the matrices. I was interested in comparing ...
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578 views

(0,1) matrix probabilities with column and row sum constraints

Given an arbitrary length set of values $c=[1, 2, 3, 4, 5]$ and a max value $m=3$, construct a random (0,1) matrix with non-zero rows such that the column sums equal $c$ and the row sums do not ...
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1answer
585 views

Distribution of Trace of non-centered Wishart matrix

I am looking for the distribution of trace of the non-central Wishart matrix with different variations along different axes. Is there a general formula for such distribution? If not, is there a ...
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133 views

Largest eigenvalue of this random 2x2 matrix

Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. ...
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350 views

Covariance of random vector multiplied with a random matrix

For a random vector $x$ multiplied by a non-random matrix $A$, $y=Ax$ the covariance matrix of $y$ is given by $\Sigma_y = E[Ax (Ax)^T] = E[Ax x^T A^T] = A E[x x^T ]A^T = A \Sigma_x A^T$, where $\...
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1answer
240 views

Why linear transformation can improve classification accuracy when the dimensionality of data is high?

Let $X$ be an $m\times n$ ($m$: number of records, and $n$: number of attributes) dataset. When the number of attributes $n$ is large and the dataset $X$ is noisy, classification gets more ...
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38 views

Statistical distribution of a scaled complex Wishart matrix

Given that $\mathbf{x}\in\mathbb{C}^{N\times 1}$, where $\mathbf{x}\sim\mathcal{CN}(0,\mathbf{R})$ then $\mathbf{x}\mathbf{x}^\mathrm{H}$ is complex Wishart, i.e, $\mathbf{x}\mathbf{x}^\mathrm{H}\sim\...
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467 views

Generate correlated multivariate normal samples with copula

I've seen examples of constructing multivariate distribution with univariate marginals coupled together via a normal copula (see Mvdc function from copula package ...
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493 views

Generating and verifying uniformly distrubuted orthogonal matrices

I would like to generate a uniformly distributed $n \times n$ orthogonal matrix. There seems to be several such methods; see this question and the oft-cited paper by Stewart. Using a QR decompostion ...
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1answer
193 views

Distribution of eigenvalues given one is known

I'm familiar with using insights from Random Matrix Theory to determine the number of principal components from the PCA of a covariance/correlation matrix to use to form factors. If the eigenvalue ...
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1answer
14k views

Expected value and variance of trace function

For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^...
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2answers
2k views

Generating random variables satisfying constraints

I need to generate a list of random variables $\bf{x}$ subject to constraints that can be expressed in the form $\bf{E}x=b$ where $\bf{E}$ is an $m \times n $ matrix if $\bf{x}$ has $n$ entries. In ...
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5answers
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Generating random matrices with sum and maximality constraints

I'd like to generate a random square matrix such that the rows are normalized to one and the diagonal elements are the maximum of their column. If there an efficient way to sample these matrices ...
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1answer
152 views

Generation of orthogonal matrices “close” to identity

Suppose I want to generate a $n \times n$ orthogonal matrix $H$ (that is, $H^T H=I$) but with the property that $1-e < (tr H)/n < 1+e$ for some pre-specified tolerance $e$. How can I do this? ...
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405 views

Generating random matrices with specific equality constraints

Suppose I want to generate a nonnegative $n \times n$ matrix $\mathbf A$ for an odd $n$ (say, $n=5$ for a good enough example), such that the individual elements are drawn from a uniform distribution ...
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1answer
1k views

Random matrices with constraints on row and column length

I need to generate random non-square matrices with $R$ rows and $C$ columns, elements randomly distributed with mean = 0, and constrained such that the length (L2 norm) of each row is $1$ and the ...
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1answer
1k views

Second moment of draws from a multivariate normal covariance matrix

Suppose I have an $N\times N$ covariance matrix that describes a multivariate normal joint distribution. Now take 100,000 draws of the covariance matrix. I measure the variance of values for each ...
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1answer
475 views

PCA with covariance matrix calculated using random matrix theory in R

I would like to perform a PCA and use the covariance matrix obtained by the random matrix theory. Is there an implementation of this in R? I am currently using the standard prcomp function from ...