Articles of random matrices

Singular vector of random Gaussian matrix

Suppose $\Omega$ is a Gaussian matrix with entries distributed i.i.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would be the distribution of the column (or row) vectors of $U$ and $V$? Would it be a Gaussian or anything closely related?

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024×1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised when I found that the eigenvalue distribution seemed to be uniformly distributed over a circular disc in the complex plane. I.e. constant density […]

How can we parametrise this matricial hypersphere?

What I call a matricial hypersphere for lack of a recognised name is the set in $\mathbb{R}^{p\times k}$ defined by $$\mathfrak{H}=\left\{ a_1,\ldots,a_k\in \mathbb{R}^{p};\ \sum_{i=1}^k a_i a_i^\text{T} = \mathbf{A} \right\}$$ where $\mathbf{A}$ is a $p\times p$ symmetric positive semi-definite matrix of rank $k$ $(k\le p)$. My questions are Is this a well-known object? Given the matrix […]

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} exp\left(-\frac{1}{2}~tr(\Sigma^{-1}HLH^{T})\right)~dH$ where ${\mathcal{O}(p)}$ is a Orthogonal group of $p \times p$ symmetric matrices, $\Sigma \neq aI $ is a covariance matrix of a column vector of a correlated Gaussian matrix $A$ that occurs in […]

Generate integer matrices with integer eigenvalues

I want to generate $500$ random integer matrices with integer eigenvalues. Thanks to this post, I know how to generate a random matrix with whole eigenvalues: Generate a diagonal matrix $D$ with the desired (integer) eigenvalues. Generate an invertible matrix $A$ of the same size as $D$. Record the matrix $A^{-1} D A$. However, the […]

Sampling $Q$ uniformly where $Q^TQ=I$

(This is related to this question) $Q \in \mathbb{R}^{n\times k}$ is a random matrix where $k<n$ and the columns of $Q$ are orthogonal (i.e. $Q^T Q = I$). To examine $E(QQ^T)$, I conducted monte carlo simulations (using matlab): [Q R] = qr(randn(n,k),0); In other words, I just sampled a $\mathbb{R}^{n\times k}$ matrix from a standard […]

Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let’s take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is equivalent of “taken a matrix at random from $M_{\mathbb{R}}(n,n)$, what is the probability that its determinant $\neq 0$) Don’t know if this question is difficult […]

If I generate a random matrix what is the probability of it to be singular?

Just a random question which came to my mind while watching a linear algebra lecture online. The lecturer said that MATLAB always generates non-singular matrices. I wish to know that in the space of random matrices, what percentage are singular? Is there any work related to this?

Why did no student get the correct answer?

I gave the following problem to students: Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$. Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same determinant and the same trace. Give an example of two $2\times 2$ […]

The use of log in the Mean density of the nontrivial zeros of the Riemann zeta function (part 2)

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: The mean density of the non-trivial zeros increases logarithmically with height $t$ up the critical line. Specifically, defining […]