Statistics of spectral properties of matrix-valued random variables.

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9
votes
2answers
781 views

Expected values of traces of products of random matrices

Suppose I want to compute a quantity of the type: $\mathbb{E}\mathrm{tr}(AUBU^{\ast})$ where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher ...
7
votes
2answers
422 views

Maximum Singular Value of a random +1/-1 matrix

Hi, Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability 0.5 to be either +1, or -1. Do you know any result in the ...
0
votes
0answers
386 views

PrincipAl Eigenvector of a Random Matrix

Let $A$ be a random matrix, let $\mathbf{x}$ be the singular vector associated with $\|A\|$. Let $\bar A$ be the entry wise expectation of $A$, and let $\mathbf{\bar x}$ be the singular vector ...
6
votes
2answers
289 views

Central limit theorem for 3d rotations

Let $X_i$ (where $i=1, \dots , n$) be independent and identically distributed 3d rotations. What is the distribution of $X_1X_2\dotsb X_n$ in the limit of large $n$? I'm especially interested in the ...
1
vote
1answer
203 views

Least singular value gaussian orthogonal ensemble.

Hello everybody, here is my question: Assume A is a random symmetric $nxn$ matrix whose entries are independent, normally distributed with mean zero and variance 2 on the diagonal and 1 off diagonal ...
1
vote
2answers
267 views

Random matrix with non-identical variances

Hello, Consider $A$ a $n \times n$ random matrix with centered Gaussian entries $A_{i,j}$ such that $$\mathbb{E}[A_{i,j}^2]=\sigma_j^2/n$$. The variances depend on the column only. What do we know ...
1
vote
0answers
143 views

Delta function representation as an integral of Pfaffians over SO(2m)

Define $\mathcal S$ as the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$. Let $S_i, S_j \in ...
2
votes
0answers
116 views

Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem. Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = ...
5
votes
1answer
383 views

Derandomizing random matrices

My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates ...
-1
votes
1answer
441 views

existence of polynomial equation system solution

For $1 \leq i \leq n$, let $A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \\ \end{bmatrix}$ $B_i=\begin{bmatrix} b_{i1} ...
2
votes
0answers
402 views

Expected operator norm of inverse Wishart matrix

Let $ W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum ...
2
votes
2answers
451 views

spectra of VERY sparse random matrices

Consider an $n\times n$ random binary matrix $M$ with i.i.d. entries $m_{ij} \sim {\rm Bernoulli}(p)$, where $p = n^{-\beta}$ with $\beta \in (1,2)$. I am interested in the behavior of the singular ...
3
votes
0answers
110 views

rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density: $p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$, is the following statement true? With ...
5
votes
1answer
783 views

Eigenvalue distributions of finite dimensional Wishart matrices

I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
5
votes
1answer
387 views

Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions

Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...
3
votes
2answers
315 views

Sample from a delta-ball in the orthogonal group O(n)

An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region. For ...
2
votes
2answers
705 views

Does the Tracy-Widom distribution describe the tails of eigenvalue densities of finite dimensional random matrices?

The Tracy-Widom distribution (TW) describes the density of the largest eigenvalue of a random Hermitian matrix, when scaled and centered appropriately (depending on GOE/GUE/GSE/Wishart, etc). In a ...
1
vote
3answers
397 views

Eigenvalues of Krylov matrices

Let an $n\times n$ matrix ${\bf A}$, the all ones vector ${\bf w}$, and the $n\times n$ Krylov matrix $${\bf K}_n = \left[ {\bf w}\;\;{\bf A}{\bf w}\;\;\ldots \;\; {\bf A}^{n-1}{\bf w}\right].$$ Is ...
7
votes
1answer
481 views

Expected norm of sum of random orthogonal matrices

Somehow I got wondering about the following question today: Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices. What is the expected value of the quantity ...
14
votes
0answers
432 views

Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
3
votes
1answer
1k views

Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
4
votes
1answer
361 views

Decomposition of Haar measure other than Hurwitz's

Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
1
vote
1answer
317 views

Eigenvalue Density of Some Random Matrices?

Consider a class of real symmetric random matrices,$M_{n\times n}=(X_{i,j})_{n\times n},$ whose off-diagonal elements follow an exchangeable distribution, and the diagonal elements follow another ...
6
votes
2answers
754 views

Statistics for Haar measure of random matrices?

Let's say I have $M$ samples of $N\times N$ real orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over ...
4
votes
2answers
1k views

Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix

TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...
7
votes
3answers
467 views

Relationship between free probability and deterministic graphs?

Consider the $N\times N$ matrix $$ M = \left(\begin{array} \\ 0 & 1 & & 0 \\ 1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ 0 & & 1 & 0 \\ ...
2
votes
4answers
2k views

Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices? My understanding for what Haar measure means for $U(1)$ is that it ...
1
vote
0answers
293 views

Relationship between R-transform and free convolution of random matrices?

I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
2
votes
4answers
433 views

Statistical computation in matrix. Rows before columns? riddle..

First I'll phrase the question as a riddle, and than as a general math problem. We have 12 lettered vases $(A,B,...,L)$, in each vase there are 30 numbered balls (1-30). In each ball there is some ...
1
vote
1answer
338 views

Random sampling a symmetric matrix

I'd like to sample the elements of a symmetric square matrix uniformly. For example, for a $N\times N$ matrix, I'd like to only keep $\alpha$% of the matrix elements to build a sparse matrix, while ...
0
votes
1answer
393 views

Spectral theory of real symmetric matrices with random diagonal elements

Can you point me in the direction of any research done on the spectral theory (i.e. eigenvalues and eigenvectors) of real symmetric matrices with random (Gaussian or Levy) diagonal elements and fixed ...
8
votes
1answer
791 views

Matrix inversion lemma with pseudoinverses

The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own. Suppose we pick $n$ values ...
19
votes
1answer
1k views

When should we expect Tracy-Widom ?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
12
votes
2answers
1k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
9
votes
1answer
553 views

Average over Random Permutations

Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the ...
6
votes
1answer
570 views

Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces?

Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. ...
8
votes
0answers
416 views

Framework for primes vs random matrices

This is inspired by What results would follow from or imply "randomness" of the primes? , but I think it is sufficiently different to ask separately. We can formalise probability in ...
0
votes
1answer
581 views

Uniform correlation matrix sampling and not so uniform laws

Hi everyone, I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways. First, I would like to determine the "uniform" distribution over the "correlation ...
4
votes
3answers
702 views

Marginal distribution of the diagonal of an inverse Wishart distributed matrix

This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here: Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
11
votes
0answers
314 views

Lower Bound on the Volume of Certain Polytopes

Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks $$ \rho=\{B_1,B_2,\ldots,B_{k}\} $$ we can define the set of equations $$ E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in ...
11
votes
1answer
739 views

A Question on Random Matrices

Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by $$ V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q}) $$ where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
3
votes
4answers
526 views

efficient way to compute the inversion of the following matrix

Hi, there I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
4
votes
2answers
272 views

analogue of GUE and Ginibre in higher dimensions

This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N ...
2
votes
1answer
238 views

Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?

It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog ...
3
votes
3answers
1k views

Distribution of trace of inverse-Wishart matrix $W_n(I,n)$

Hello, I'm interested in the distribution of the trace of an inverse-Wishart matrix $W_n^{-1}(I,n)$, where $I$ is $n\times n$ identity matrix. More precisely, I seek for an asymptotic estimate (when ...
3
votes
0answers
231 views

Stable distributions for Lindeberg exchange strategy?

Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
4
votes
2answers
568 views

Distribution of eigenvalue spacings

I have been doing some experiments on classes of random matrices, and it seems (visually) that the distribution of eigenvalue spacings is consistent with GOE or GUE or GSE. Unfortunately, to test ...
8
votes
0answers
341 views

Has the technique of “sprinkling” been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
11
votes
4answers
848 views

Why only three classical matrix ensembles in RMT? (Newbie question)

I am just starting out on understanding random matrix theory from a background in applied mathematics. I have a very basic question about the Gaussian ensembles: why are there only three classical ...
4
votes
2answers
487 views

Induced p-norm of a Random matrix

This question is related to my earlier question here . Given an $n\times n$ random matrix $A$, is determining the properties (mean, variance,moments,etc.) of its induced $p$-norm ($p\neq ...