For questions concerning random matrics.

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Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
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0answers
40 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
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197 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
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0answers
37 views

Derivative of Cholesky decomposition

I would like to compute the derivative of the Cholesky decomposition, for example I have a matrix 2 x 2 R = 1 rho rho 1 where rho is a parameter, now I compute the Cholesky decomposition of ...
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2answers
46 views

Constructing 5 by 5 Unitary matrices

I am trying to construct an arbitrary 5 x 5 Unitary matrix. Any example will be appreciated.
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0answers
25 views

Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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1answer
19 views

Independence of distribution

Let there be a random matrix defined as $\mathbf{H}_1 = X + \boldsymbol\nu$, where, $X$ is deterministic and $\boldsymbol\nu$ is Gaussian white noise. Now let there be another random matrix defined as ...
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1answer
99 views

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)} ...
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0answers
23 views

radius of convergence of hypergeometric functions

Hypergeometric function of scalar arguments is defined as \begin{eqnarray} _aF_b\left(p_1,...,p_a;q_1,...,q_b;z\right) &=&\sum_{i=0}^{\infty} ...
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1answer
139 views

Why does this determinant have a continuous density at zero?

This question is a simplification of my previous question. I think this is easy, but I don't have a strong enough background in probability. Let $A$ be a random $n\times n$ real matrix that satisfies ...
2
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1answer
33 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
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1answer
93 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
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24 views

Projection and matrix norm

Suppose we are in the matrix space $\mathbf{R}^{n_1 \times n_2}$. Suppose, $R_{\Omega}$ is an operator, such that $R_{\Omega}(Z)$ chooses $m$ entries from $Z$ uniformly at random with replacement and ...
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1answer
50 views

Important topics in Matrix analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
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0answers
47 views

Plotting the pair correlation function for the zeta zeros /GUE

I am making a shameless request for instructions on how to plot this: from this page. I can see from here that normalizing the zeros is given by ...
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0answers
30 views

Canonical Form of Nilpotent Matrices

Given the matrix $$\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$ where $S$ is an $n \times n$ ...
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0answers
48 views

Bounding the spectral norm of a block random matrix

Suppose that zero-mean iid random matrices $A_1 ,A_2,\dotsc,A_n$ satisfy $$\mathbb{P}\left(\left\|A_i \right\|\geq t\right)\leq \phi\left(t\right),\tag{*}$$ for $t>0$, where ...
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0answers
27 views

Distrbution of Matrix times vector

Given the distribution of an $n\times n$ matrix $A$, how to find the distribution of $Y = AX$, where $Y$ and $X$ are $n\times 1$ vector and X is deterministic.
5
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1answer
82 views

Closed formula for mean

Suppose we have the i.i.d. random variables $X_{11}, X_{12},\ldots, X_{nn}$, such that each $X_{ij}$ has standard normal distribution $N(0,1)$, with mean $0$ and variance $1$. Given some integer ...
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3answers
43 views

Random matrices in coordinate independent way

How to generate a random matrix in a basis independent way (so that the random distribution does not change if the coordinates are rotated)? I am especially interested in generating random rotation ...
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1answer
55 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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0answers
24 views

Distance of random span to a vector

I've been batteling with the following problem: Assume we have a diagonal matrix $D \in \mathbb{R}^{l \times l}$, a vector $\beta \in \mathbb{R}^l$. Next we simulate a random matrix (Idea inspired by ...
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1answer
38 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
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0answers
44 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
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1answer
38 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...
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0answers
36 views

Expectation of the “second eigenvalue” of a random binary matrix

For a real matrix $A$, define its second eigenvalue to be $\max_{i\ge 2}|\lambda_i|$, where $\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_n$ are the eigenvalues of $A$. What is the expectation of the ...
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0answers
224 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
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0answers
26 views

Maximal area of a monochromatic combinatorial rectangle

I'm stuck on this one and would appreciate any help: Let $M$ be a $2^n \times 2^n$random matrix of $0$'s and $1$'s (for each entry, there is probability of $\frac{1}{2}$ that it will be $0$, same ...
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0answers
111 views

What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
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1answer
64 views

Random operators

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
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1answer
76 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
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0answers
15 views

Expected products of entries in a random unistochastic matrix

Consider the "unistochastic" distribution over $m \times m$ doubly-stochastic matrices obtained by taking the norm-squared of entries of a Haar-random unitary matrix. I'd like to know the expected ...
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0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
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2answers
89 views

Eigenvalues of a Random Matrix

I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you. A random matrix is defined as a matrix whose entries ...
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1answer
107 views

Random Matrix Theory and ESD

I need some help to understand what professor Terence Tao means in this part of "Topics in random matrix theory". I'm having a hard time to undertand this function ESD. How is possible for it to be ...
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0answers
39 views

Entries of a Haar distributed unitary matrix

The eigenvector matrix of a Wishart matrix is Haar distributed and that implies that the eigenvectors are uniformly distributed on a sphere. I'm interested to know what is the distribution of ...
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0answers
50 views

Eigenvalue distribution of sum of random matrices

Suppose we have a set of freely independent random matrices $X_i$ $i=\{1,\dots,n\}$ of dimension $M\times N$. Each $X_i$ has independent and identically distributed entries. Then we make the sum ...
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20 views

Are these two matrices asymptotically free?

The entries of $\mathbf{H}_1\in\mathbb{C}^{K\times M},\mathbf{H}_2\in\mathbb{C}^{K\times M},K<M$ are all i.i.d. RVs $\sim\mathcal{CN}(0,1)$. $\mathbf{D}\in\mathbb{R}^{K\times K}$ is a deterministic ...
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0answers
92 views

correlation matrix of the random matrix

$\mathbf{X}$ is random matrix given by $\mathbf{X}=\left[\begin{array}{*{20}{c}} \mathbf{x}_{1}&{}&{}&{}\\ {}&\mathbf{x}_{2}&{}&{}\\ {}&{}& \ddots &{}\\ ...
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1answer
71 views

What is the eigenvalue of matrix from matrix minus all-ones matrix?

Suppose we know the eigenvalues of matrix A, and J is all-ones matrix with all elements are one. Then what are the eigenvalues of A-J? ps. A is random matrix with element from distribution N(0,s) ...
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0answers
60 views

Problem about the determinant of a random matrix

I am being haunted about this problem on the value of the determinent of this Random Matrix ever since it came into my mind last week.The problem goes like this: Suppose $A$ is a square matrix of ...
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3answers
104 views

How sum work, vectors and matrices

I am having troubles understanding what is going on here. Would anyone be able to do this step by step with values so that I will be able to understand the SUM and how it works. xi is in this case ...
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0answers
36 views

Concentration Inequalities For Matrices? (Around Mean)

I need a result which talks about concentration of a 'random matrix' around its expected value. I need the following: $Pr(||X-E[X]|| \le \epsilon) \ge \hspace{2pt} ? \hspace{8pt}, X \in \mathbb{R}^{n ...
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2answers
99 views

Find the Norm of Matrix using Cauchy-Schwarz inequality

Let $A$ be $n \times n$ matrix and such that all of its entries are uniformly $O(1)$. Using Cauchy-Schwarz inequality, show that the operator norm of matrix $A$, which is $\|A\|_{op}=\sup_{x\in R^n: ...
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1answer
62 views

Relation between the eigenvalue density and the resolvent

Many texts (e.g. 1-2-3) on random matrices start with some variation of the identity: $$\rho_1(\lambda) = \frac{1}{\pi} \text{Im}\{\langle\text{Tr}(\mathbf{X}-\lambda\mathbf{I})^{-1}\rangle\}$$ ...
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1answer
114 views

Obtaining the Airy kernel from the Christoffel-Darboux formula with asymptotic Hermite polynomials

Let the Kernel associated to a family of orthogonal polynomial $p_n(x)$ with weight $w(x)$ be defined as $$K_N(x,y):=\frac{\sqrt{w(x)w(y)}}{\int w(x) p_{N-1}(x)p_{N-1}(x)dx} ...
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1answer
211 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
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0answers
67 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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0answers
38 views

The differentiation of the trace of complex matrix

Condition: all the matrices are complex. $\dagger$ denotes the conjugate transpose, $*$ denotes the conjugate, $\mathop{Trace}$ denote the trace of a matrix. What is the differentiations of the ...
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1answer
65 views

Calculating the Lyapunov index of elements of a matrix.

In Random Dynamical Systems - Arnold Ludwig, Example 3.3.9. The cocycle generated by a random matrix: $$A=\left(\begin{array}{cc} a & c\\ 0 & b \end{array}\right)$$ is given by: ...