For questions concerning random matrices.

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16
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6answers
2k views

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 ...
1
vote
0answers
15 views

Finding “the” Marchenko-Pastur distribution in the original article of 1967

I am looking at distribution properties of eigenvalues of sample covariance matrices. Following the Wikipedia article on the Marchenko-Pastur distribution: Let $X$ denote a $M \times N$ random ...
1
vote
0answers
22 views

Find a hypergeometric formula embracing three specific cases

For a parameter value $a=\frac{1}{4}$, I have the result \begin{equation} Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}...
0
votes
0answers
14 views

What is Steiltjes transform (other Integral transform) and how does it helps in probability theory, specifically in random matrix theory?

I have started growing interest in random matrix theory. Trying to understand it from "Random Matrices" by Madan Lal Mehta and "An Introduction to Random Matrices" by Anderson and many sources on ...
0
votes
0answers
19 views

Intuitive explanation for Marcenko-Pastur law

I am looking for an intuitive reasoning behind the Marcenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
1
vote
0answers
45 views

closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
1
vote
2answers
54 views

Equivalent Definitions of the Spectral Norm

There are many equivalent definitions of the spectral norm $\|A\|_2$ for when $A$ is a symmetric matrix, the most common ones being $$\sup_{\|x\|_{2} = 1}{\|Ax\|_{2}} = \sup_{\|x\|_{2}=1}|{\langle Ax,...
1
vote
0answers
22 views

Construct a master (possibly hypergeometric) formula from a family of formulas indexed by the half-integers and integers

I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a ...
0
votes
0answers
57 views

The random matrix for Riemann Hypothesis, is it corresponding to an operator in quantum mechanics or in quantum field theory?

Odlyzko showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This ...
0
votes
0answers
10 views

Wishart plus scalar multiple of identity

Is the sum of a Wishart distributed matrix and a scalar multiple of identity matrix, another Wishart distributed matrix? I guess it is not. If not, what is the distribution called and can its density ...
0
votes
0answers
18 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
-3
votes
3answers
156 views

how can I create a random matrix with specific entries

I would like to create/generate a random square $n \times n$ matrix with the following specifications: the first and the last row of the matrix are nonzeros likewise nonzero at the main diagonal ...
1
vote
0answers
41 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
0
votes
0answers
35 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
0
votes
0answers
13 views

Determine eigenvalue distribution support

I am working on a project regarding random matrix spectra and I need some help with the following: let us assume we are looking at some particular family of NxN random matrices in the limit of N -> ...
0
votes
2answers
33 views

Proving $(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$ , $P$ idempotent matrix.

Given that a matrix $P$ is idempotent how to prove the following relation: $$(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$$ $c$ is any real constant.
0
votes
0answers
24 views

Mock gamma factors

I wonder if analogues of gamma factors as used to defined a complete L-function of the form $\displaystyle{\prod_{j=1}^{\infty}\Gamma(\lambda_{j}s+\mu_{j})}$ with a possibly non integral value of $d=\...
0
votes
0answers
18 views

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 $$ P(s) = \frac{\pi}{2}s e^{\frac{-\pi s^2}{4}} $$ ...
3
votes
1answer
32 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 ...
0
votes
0answers
11 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 condition)....
3
votes
1answer
60 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 ...
0
votes
0answers
15 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?
0
votes
0answers
39 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 \{1,\...
2
votes
2answers
31 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\\ AQ=...
0
votes
0answers
26 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 $a_1,\...
0
votes
0answers
9 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 ...
1
vote
2answers
35 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: ...
1
vote
0answers
37 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 ...
1
vote
1answer
28 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 $X$....
3
votes
0answers
32 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 ...
0
votes
1answer
20 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 $V$ ...
6
votes
1answer
127 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 ...
0
votes
0answers
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 quantum-...
1
vote
1answer
40 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 ...
0
votes
0answers
44 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 $U_{(\...
0
votes
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} ...
1
vote
0answers
33 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 ...
0
votes
0answers
26 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 ...
0
votes
1answer
24 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 N_p(0,V)...
0
votes
0answers
23 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 $R\in\mathbb{R}^{p\...
0
votes
0answers
10 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 ...
2
votes
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 $M^\...
1
vote
1answer
28 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 ...
1
vote
0answers
46 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 ...
3
votes
0answers
92 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 ...
14
votes
1answer
241 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 ...
1
vote
1answer
40 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 ...
2
votes
1answer
106 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 ...
0
votes
0answers
17 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 $P_{n\...
5
votes
3answers
223 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 $$\mathbb{E}(\log(\det(AA^...