Statistics of spectral properties of matrix-valued random variables.

**9**

votes

**2**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**3**answers

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

**4**answers

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

**0**answers

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

**4**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**1**answer

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

**4**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**4**answers

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

**2**answers

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 ...