For questions concerning random matrices.

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Assumptions that led to Wigner's surmise for probability density of spacing between eigenvalues of real symmetric random matrices.

In 1957, Wigner surmised (guessed) that the probability density of the spacing between adjacent eigenvalues of real symmetric matrices is given by $$ P(s) = \frac{\pi}{2}s e^{\frac{-\pi s^2}{4}} $$ ...
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1answer
26 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
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0answers
9 views

Resolvent of Random Matrix is N^2 Lipschitz

For $\tau > 0$, let S($\tau$) = {E + i$\eta$ | $\tau^{-1} \leq E \leq \tau$, $\eta \geq N^{-1 + \tau}$}, and let $H$ be an $N\times N$ Wigner random matrix (i.i.d entries up to Hermitian ...
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1answer
54 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the functiona ...
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11 views

is it possible to generate random ill-conditioned sparse matrix in MATLAB?

I know how to generate ill-conditioned dense matrices (use 'gallery'). I do not know how to generate ill-conditioned sparse matrices. Anyone help?
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38 views

probability of a random vector in row space of a random matrix

Suppose we have a random matrix $A$ of dimension $n\times m$ (let $m<n$) with entries in $F_2$ ( each entry in $A$ is 0/1 with probability 1/2). Suppose I fix a $x\in \{0,1\}^m$ and $k\in ...
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2answers
30 views

Under what condition on matrix $Q$ we have $tr(AQ)=tr(BQ)$

Let $A,B$ are similar matrices. Then, under what condition on matrix $Q$, we have $tr(AQ)=tr(BQ)$ $A$ and $B$ are similar matrices, so there exist an invertible matrix $P$ such that $$A=P^{-1}BP\\ ...
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22 views

Linear independent in random variable and observations

I am confused with some fundamental concepts. Here for $n$ random variable $X_1,\cdots,X_n$, i.i.d and follow standard normal distribution, the probability that there exists a set of constant ...
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6 views

Deriving spectral norm or similar quantity for structured random matrix

I have a problem where I have no idea to start. Suppose a simple Least Squares system with $M$ unknowns $c$ and $N$ observations $y$ which is given through the linear mapping $X$: $$y = X c$$ It is ...
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2answers
14 views

Set density of random matrix in Sage

I'm using Sage to calculate a bunch of matrix operations over GF(2), using the code below to randomly generate an invertible matrix: ...
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0answers
28 views

Probability of n distinct eigenvalues

For a randomly generated $n$ by $n$ matrix, is the probability that it has $n$ distinct eigenvalues equal to $1$? I have a feeling it must be. But, if that's the case, why do we concern ourselves so ...
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1answer
22 views

Distribution of $aXa^T$ for normal distributed vector $a$

Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix ...
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0answers
26 views

The distribution of the condition number for complex Wishart matrices.

I am trying to derive the distribution of the condition number for centered uncorrelated complex Wishart matrices $n\times n$ with $m$ degrees of freedom. The problem is with the solution I got (it's ...
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1answer
18 views

Generate a random binary full-rank rectangle matrix that is a basis of a subspace

Disclaimer: I think of vectors as row vectors. I have a full-rank $m \times n$ ($m < n$) binary matrix $B$ which is a basis of $m$-dimensional subspace $V \subset\mathbb F_2^n$ (i.e. subspace $V$ ...
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1answer
126 views

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
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10 views

Reconstruct a family of probability distributions having certain generalized hypergeometric moments

Reconstruct and/or otherwise characterize any/or all members of a certain one-parameter ($\alpha =\frac{1}{2}, 1, \frac{3}{2}, 2,\ldots$) family of univariate probability distributions (of ...
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1answer
38 views

Exercise 12 in Tao's notes on the semi-circular law

My question concerns this exercise in Tao's notes on the semi-circular law for random matrices. We are trying to solve equation (22) in the notes, which is a quadratic equation whose coefficients are ...
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0answers
42 views

Sequence of infinite Backward Products

I am not a mathematician but need to understand an asymptotic property of Backward Products of row-stochastic matrices. Let's say I have the following $\textit{sequence}$ of backward products ...
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0answers
15 views

Explanation of a kind of spectral measure.

Could someone help me, please, to understand in term of values of a Matrix $M=(m_{i,j})_{i,j\in\{1,n\}^2}$ the following measure : $$ \frac1{n} \sum_{i=1}^n \langle v_i,e_j \rangle \delta_{\lambda_i} ...
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0answers
32 views

Generalizing lemma 1 in Tao's notes on operator norms of random matrices

My question concerns the proof of Lemma 1 in this blog post of Terence Tao. In the first paragraph, he says: Applying standard concentration of measure results (e.g. Exercise 4, Exercise 5, or ...
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22 views

Construction of matrices with special property

Can any one help me in constructing two matrices $A$ and $B$ (can be singular) such that $R(A)\subseteq R(B)$ and $R(B^*)\subseteq R(A^*)$. Here $A^*$ means conjugate transpose of $A$. It will be ...
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7 views

Gaussian expected inner product with respect to fixed matrix

Suppose $R\in\mathbb{R}^{p\times p}$ is a a fixed matrix (it can be asymmetric, non-positive definite, and so on). Then, I would like to find a formula for the expectation $$\mathbb{E}_{x\sim ...
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0answers
16 views

Expectation over Wishart distribution of rotated trace

Suppose $S$ is drawn from the Wishart distribution $W_p(V,n)$ with $n$ degrees of freedom and positive definite scale matrix $V\in\mathbb{R}^{p\times p}$. Then, given a matrix ...
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7 views

Matrix ensemble for complex symmetric matrices?

In random matrix theory, two of the main Gaussian ensembles are for real symmetric (GOE) and hermitian (GUE), both which come from taking a matrix of i.i.d real (respectively complex) Gaussian random ...
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1answer
21 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij} \overset{\text{iid}} \sim N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of ...
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1answer
21 views

Understanding the proof of Johnson-Lindenstrauss (JL) lemma

I have some questions about proof of Johnson-Lindenstrauss (JL) lemma. I appreciate any responses in advance. It is stated in the following paper: An elementary proof of JL lemma we argued: "Hence ...
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0answers
37 views

Necessary/Sufficient conditions eigenvalues random matrix

I am working on the computational analysis of the eigenvalue spectra of real symmetric random matrices. At the moment my approach is fully empirical but I would like to develop a rough model of the ...
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0answers
82 views

Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
13
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1answer
225 views

Expected value of the smallest eigenvalue

Consider a random $m\times n$ matrix $M$ with elements from $\{-1,1\}$ and $m<n$. What is known about the expected value of the smallest eigenvalue of $MM^T$? The following picture shows ...
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1answer
38 views

Numerical diagonalization of a random hermitian matrix $H=U\Lambda U^{-1}$: enforce uniqueness and uniformity of $U$

I've stumbled across this seemingly simple question, but I could not find a satisfactory answer. Suppose I have a complex hermitian random matrix $H$. It can be diagonalized by a unitary ...
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1answer
103 views

Eigenvalues of $AA^T$ for random circulant $A$

Consider a random circulant $n$ by $n$ 0-1 matrix $A$. Let $P(A_{1,i} = 1) = 1/\sqrt{n}$ and all the elements of the first row be independent. We know that the expected value of the diagonal ...
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16 views

Singular values after projection

Suppose we have a random matrix $\Phi$ where entries of $\Phi$ are i.i.d random variables and we would like to construct a new matrix $\Gamma$ in the following way: Compute $P\phi_i$ where ...
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3answers
217 views

Expected value of $\log(\det(AA^T))$

Consider uniform random $n$ by $n$ matrix $A$ where $A_{i,j} \in \{-1,1\}$. We know that with high probability $A$ is non-singular. Are there known estimates or bounds for ...
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1answer
30 views

Eigenvectors of Diagonally Dominant Matrices

Given a diagonally dominant positive semi definite matrix $A \in R^{n\times n}$, with eigen decomposition $A = U\Sigma U^T$, can we say anything about the form of $U$. For example, to sample ...
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1answer
90 views

Probability of having complex eigenvalue?

Let $A$ be a real $n \times n$ matrix with coefficients randomly chosen from Uniform Distribution [-1,1]. What's the probability that A has a complex eigenvalue with non-zero imaginary part? If you'd ...
3
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1answer
55 views

Distribution of matrix eigenvalues

I would appreciate if one could help me figure out this problem. I have a matrix $G$ (for simplicity assume square matrix $n\times n$). I know that if I multiply $G$ with a unitary matrix $U$ as ($A ...
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1answer
30 views

expected value of multiplication of random matrices [closed]

Suppose $X\in R^{n \times n}$ is a symmetric positive definite random matrix and $A\in R^{n \times n}$ is a symmetric constant matrix. how can I compute the below expression $$ E[X^TAX] $$
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0answers
27 views

expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution. Background The Wishart distribution with $\nu$ degrees of freedom and positive definite ...
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0answers
22 views

Difference between the SCM converging to the Marcenko-Pastur distribution and Johnstone's result about the top eigenvalue

I have a confusion which I suppose must be rather basic. As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard ...
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0answers
38 views

Why is Gaussian matrix full rank?

Suppose $A\in R^{n\times n}$ is a matrix with independent standard normal entries. Is there an elementary argument to show that $A$ is nonsingular with probability $1$?
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1answer
28 views

How to prove $E\|Y'\|\leq E\|Y'-Y''\|,$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$?

How to prove $$E\|Y'\|\leq E\|Y'-Y''\|,$$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$; $\|\cdot\|$ denotes the $l_2$ operator norm;$E$ ...
2
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0answers
37 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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12 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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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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44 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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11 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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0answers
32 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
38 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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1answer
47 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
53 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 ...