For questions concerning random matrices.

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17 views

Expectation of the matrix product $Q A (Q A)^h$

Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$. We define $A$ as an $n \times k$ ...
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10 views

creating matrix normal distribution

would you please help me? I have a distribution on $vec(A)$ as below $$ q(vec(A))=N_{np}(A|vec(\mu),Q) $$ In which $N(.)$ means the normal distribution and $$ Q=L\otimes K+ H\otimes J $$ How can ...
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14 views

Exercise 2.1.5 in An Introduction to Random Matrices by Zeitouni et al.

I have a question regarding exercise 2.1.5 on page 19 in this book: http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf I would like a reference or help on this exercise. The exercise asks the ...
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0answers
30 views

Invertibility, inverse, and line weight of big circulant matrices

I am generating a random square sparse binary circulant matrix, defined by its first row. The length of the matrix is 9857 bits, and each line contains 71 ones, the rest are zeroes. I need to ensure ...
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8 views

Inequality in the proof of a LDP for the largest eigenvalue of the GOE

In the proof of theorem 6.2 in Ben Arous' paper (page 48ff) one wants to get an upper bound on $\sigma^N\left(\lambda_N^*\geq x, \max_i^N|\lambda_i|\leq M\right)$, where ...
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16 views

Expected number of binary random vectors for dimension reduction

Assume that Trent has a binary random vector generator which creates vectors of length $n$. Each element of these vectors can be either zero or one with equal probability. Trent creates a set of $M$ ...
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1answer
34 views

Derive formula for Gaussian distribution of a matrix variable

Let $J$ be a random matrix, i.e. all elements are drawn randomly, with zero mean and $\mathrm{E}[J_{ij}^2] = \frac{1}{N}$ and $\mathrm{E}[J_{ij}J_{ji}] = \frac{\tau}{N}$, then it is given that its ...
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25 views

Almost sure convergence of eigenvectors under nice assumptions

I have that a sequence of symmetric, real, positive semi-definite random matrices, $M_n$, converges almost surely to a real-valued positive semi-definite diagonal matrix, $D$, with at least one ...
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1answer
33 views

Understanding the meaning of seed in generating random values?

I am working on a project and I'm reading this description on generating random bits in a file. They use the word "seed" in it. I've read what seed does, but I'm not quite sure how to apply it in this ...
2
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1answer
33 views

Inequality on the trace of the resolvent of a matrix

For a (random) hermitian matrix $M$ and a complex $z$, it is well known that $$ \left| \int_{\mathbb{R}} \frac{1}{z-x} \text{d}\mu_M(x) \right| = \left| \frac{1}{n} \text{Tr} (z-M)^{-1} \right| \leq ...
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27 views

Solve discretized Marcenko-Pastur equation analytically

Is there a way to solve the below complex polynomial equation analytically? Note $z \in \mathbb C^+ $ (upper half of $\mathbb C$-plane) $, n, p \in \mathbb Z^+, t_i \in \mathbb R, i=1,...,p$: $$m(z) ...
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24 views

Is exponential of GUE random matrix Haar random?

Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices. I would expect that for large $t$, the resulting measure on the unitary ...
2
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53 views

Conditions for convergence to Gaussian distribution

Let $$n_i(t)= H(u_i(t))$$ where $N\geq i\geq 1$, $H(.)$ is the Heaviside function and $$ u_i(t) = \sum_{j=1}^N J_{ij} n_j(t) $$ We start with a random $\vec{n}(0)$ and each step of time ...
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1answer
24 views

Asymptotic Distribution of eigenvalues and eigenvectors

Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ be a random matrix. Now consider the following two cases: Suppose $n$ is fixed and is a small number less than 100. If I take sufficiently large samples, N, ...
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50 views

Random matrices over $\mathbb{Z}_q$ — Reference

I've found several papers studying random matrices over finite fields $\mathbb{F}_q$, but none dealing with matrices over the rings $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$ for general $q$. Does ...
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17 views

Integrating Wishart density

I have several points $\textbf{s} = s_1,...,s_n$ which follow Wishart distribution. In one of my problem, I have to integrate this Wishart pdf over a ball of radius $r$ at origin in $\mathbf{R}^2$ ...
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26 views

Fast computation of component-wise $\exp(-XY^T)G$ for random $G$

I have the following question: Suppose I have two matrices $X,Y$ both of size $m\times p$ and a random i.i.d Gaussian matrix $G$ of size $m \times k$, $m\gg p>k$. Is there a fast way to compute ...
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19 views

About the Alon-Krivilevich-Vu result on concentration of eigenvalues of random matrices.

I am looking at the statement in theorem 5.4 in these notes, http://www.math.ucla.edu/~nickcook/talagrand.pdf I had a few questions about the statement of this theorem, Can someone kindly clarify ...
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1answer
384 views

How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. To be clear, $A$ is a matrix whose entries are chosen from $\{1,-1\}$. Let $B = A^T A$. We ...
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6 views

Dependence of left-singular vectors of random matrices

Let $h_1, h_2, h_3$ be random, i.i.d. normally distributed vectors in $\mathbb{C}^{m}$ (each vector has i.i.d. entries, and vectors themselves are independent of each other). Define the matrices $H_1 ...
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12 views

Random matrix on Clifford algebra within a specific grade

There has been some discussions about random matrices on generic Clifford algebra arXiv:1312.6291. However I would like to consider a more specific case by restricting the random matrix within a ...
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1answer
41 views

Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions: ...
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36 views

Rank of a random matix

This arises in Time-Series modelling. Suppose $Y_i \sim N_p(0,\Sigma_i)$ and they are not necessarily independent (but assuming $\Sigma_i$ to be p.d.). Then for any ${a}\neq 0 \:\:\:\:$ $Y_i'a\neq0$ ...
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1answer
37 views

An inequality in introduction to Random Matrix theory.

I am stumped in some silly easy inequality, which I don't understnad why I don't get it's on page 11 of the book by Zeitouni, et al: http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf I am ...
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1answer
73 views

Is the sum of a Wishart matrix and a deterministic psd matrix “almost Wishart”?

Let $XX^T$ be a Wishart matrix, generated by taking the columns of $X$ to be i.i.d. standard $p$-variate normal vectors. Let $AA^T$ be a non-random positive definite matrix. Though it is not possible ...
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0answers
15 views

What is the support of the PDF of eigenvalues of a random matrix called?

I only have a pedestrian familiarity with the study of random matrices, but is there a concise way to refer to the set of possible eigenvalues a random matrix can achieve? For example, given a real ...
2
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2answers
50 views

$(2\pi)^{-n/2}\int_{\mathbb{R}^n} q(x)e^{-\Vert x \Vert^2/2}\,dx = \mbox{trace}(Q)$

Let $Q$ be a symmetric matrix and consider the quadratic form $q: \mathbb{R}^n \rightarrow \mathbb{R}$, $q(x) = \langle Qx, x \rangle$ Show that $(2\pi)^{-n/2}\int_{\mathbb{R}^n} q(x)e^{-\Vert x ...
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58 views

Eigenvalues of a simple random matrix (Girko's circular law?)

I am considering a real random matrix whose off-diagonal entries are i.i.d. and normally distributed (with non-zero mean and non-unit variance), and whose diagonal entries are uniform and ...
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34 views

Any reference for the following Hadamard property

Let $x \in \mathbb{C}^{N \times 1}$ be a vector of random variables with $x \sim \mathcal{N}_C (0, \Sigma_{xx})$ and let $A \in \mathbb{C}^{N \times N} $ be a Hermitian matrix (i.e. $A = A^H$ with ...
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1answer
53 views

How to show the quotient of dual space and the annihilator of its subspace is equal to dual of the subspace? [duplicate]

Let $W$ be a subspace of a vector space $V$ over a field $F$. Let $i : W \to V $ denote the inclusion map. Show that $ \pi: V^*\to W^*$given by $\pi(f) = f\circ i$ is a surjective linear map, with ...
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2answers
40 views

How to check whether a polynomial is annihilating polynomial?

When I what to find the minimal polynomial of a matrix, first I want to get the characteristic polynomial of the matrix. But how can I check whether some polynomial is annihilating polynomial to be ...
2
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1answer
35 views

Convergence in probability of random probability measures

I was looking at the proof of Wigner's Semicircle Law from this book.. Let $L_N$ be a random probability measure and $\sigma$ be a known probability measure supported in the interval $[-B,B]$. We ...
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2answers
49 views

Random complex orthogonal matrices

How can I uniformly extract a random complex orthogonal matrix $\Omega\in O(3,\mathbb{C})$? It is easily found in the literature the uniform measure for unitary and real matrices, but I couldn't find ...
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0answers
31 views

The rank of the sum of outer products distributed according to a Haar measure

While working on the journal version of our paper, we encountered the following problem, which seems to be fairly simple but we could not find the answer: Suppose $A_i \in \mathbb{R}^{p\times n}$ are ...
6
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1answer
87 views

Minkowski-like inequality for the trace of outer products of random vectors

I am wondering if the following inequality is correct and can be shown? Let $A$ and $B$ be random vectors of dimension $n$. Then for $ p \ge 1$ \begin{align} E^{\frac{1}{2p}} \left[ \left| Tr ...
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1answer
44 views

Expected value of random matrix

Let $\mathbf{A}$ be a random matrix and $\mathbf{B}$ a matrix composed of constants. Is it possible to take $\mathbf{B}$ out of the expected value: $E\{\mathbf{A}^H\mathbf{BA}\}$. I tried with ...
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0answers
80 views

Sum of eigenvalues and relation with the matrix rank

Consider a matrix $\mathbf{H}\in\mathbb{C}^{N\times M}$ whose elements $h_{kl}$ $(k = 1, \dots, N$;$l = 1, \dots, M)$ are i.i.d. complex-valued random variables following the normal distribution with ...
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30 views

Big oh pee notation for random matrices

Is there any consensus on what $A_n = O_p(n^{-1})$ means for a sequence of random matrices $\{A_n\}_{n\geq 1} \in \mathbb{R}^{d\times d}$? Does it mean that the operator (spectral) norm of $A_n$ ...
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0answers
33 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
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1answer
63 views

Random matrix theory

A random symmetric $2 \times 2$ matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22}\end{pmatrix}$ is a member of the gaussian orthogonal ensemble (GOE), if it satisfies three ...
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14 views

Realization of Gaussian Unitary Ensemble (GUE) on [0 x 1] x [0 x 1] Grid.

I can generate Gaussian Unitary Ensemble (GUE) using the following code: ...
2
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1answer
293 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
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0answers
39 views

Can any random matrix be decomposed as a product of its covariance matrix and matrix of iid entries?

I've read some random matrix papers and the authors usually considered an $n \times p$ matrix $X$ which is defined to be $X := Z \Sigma^{1/2} $, where $Z$ is a matrix whose entries $Z_{ij}$ are i.i.d. ...
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1answer
44 views

Let $A$ be a random matrix with i.i.d entries, what can we say about $Ax$?

Assume $A$ is an $m\times n$ random matrix with i.i.d entries, and $x\in\mathbb{R}^n$ be a fixed vector with $\Vert x\Vert_2=1$. Then can we say something about $y:=Ax$? Does $y$ still have i.i.d ...
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2answers
65 views

Generating a random matrix with all eigenvalues equal to one

I want to generate a random matrix whose all eigenvalues are equal to one. How can I do it? I know one method to generate matrices with given eigenvalues is to generate a random orthogonal matrix $Q$ ...
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0answers
59 views

Symplectic Eigenvalues of Wishart Matrix

We work over the reals. Fix a dimension $n$ and a symplectic form $\Omega$. This is a $2n \times 2n$ matrix s.t. $\Omega^2 = -1$. A symplectic matrix is a matrix $S$ such that $ S^T \Omega S = ...
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1answer
31 views

Is there a way to generate random matrix that satisfying a give Linear Matrix Inequality?

The title is self-clear, I think. For example, we could generate a $n \times n$ random rank 1 positive semidefinite matrix by generate a random vector $x$ and $x'x$ will be the random matrix we want. ...
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61 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
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3answers
42 views

$A$ is a doubly stochastic matrix, how about $A^TA$

I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix. However, the paper says $A^TA$ is symmetric and stochastic. Since $A^TA$ is symmetric, if $A^TA$ is ...
4
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2answers
186 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with a normal distribution with the mean 0 and variance $m$) to its diagonal. A is a square ...