Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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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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 example, I generate upper triangular matrix, and fill…
Haitao Du
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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 zero mean, and constrained such that the length ($L_2$ norm) of each row is $1$ and the length of each column is $\sqrt{\frac{R}{C}}$.…
Tyler Streeter
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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 techniques in order to work with random matrices?…
Eduardo
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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, $Var[Tr(X^TAX)]$? Please note that $A$ is not a random…
hearse
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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 all the cases I'm dealing with, $n >> m$, for…
Mike Flynn
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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 associated with the first PC is large, then it…
John
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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 $\mathbb{R}^2$ the points could be placed on a…
Cesare
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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 but with the equality constraints the sum of…
Sami Liedes
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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 uniformly? $2 \times 2$ matrices are straightforward…
Blake Riley
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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 of eigenvalues values of a large Wishart matrix…
michek
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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 general formula for the distribution of eigenvalues of…
Andrei Kucharavy
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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, but I imagine one could use random matrix theory…
Hooked
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PCA: inference on the proportion of explained variance, in a large p setting

I am interested in doing inference on the proportion of total variance explained by the first principal component, for a PCA based on the correlation matrix R. I want to know the (asymptotic) distribution…
Matifou
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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 $\Sigma \sim W^{-1}(\nu, \Psi)$. Also let $A$ be a $q \times p$ matrix.…
jds
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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 $\mathbf{X} = (X_1,\ldots,X_n)^\top$ which follows a multivariate Gaussian…
Orlando
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