Statistics of spectral properties of matrix-valued random variables.

**27**

votes

**2**answers

1k views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...

**0**

votes

**0**answers

98 views

### Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?

Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...

**15**

votes

**0**answers

772 views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**2**

votes

**0**answers

774 views

### Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...

**0**

votes

**1**answer

239 views

### distinguishing random orthogonal matrix from Gaussian random matrix

Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ ...

**9**

votes

**2**answers

195 views

### Iterating Random Matrix Operations

Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...

**1**

vote

**0**answers

180 views

### Generalization of Lagrange inversion with “skewed” formal parameter

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ...

**1**

vote

**1**answer

172 views

### random matrix products reference

For a long time the standard (though not the easiest to find) reference on random matrix products was Bougerol and Lacrois:
Bougerol, Philippe, and Jean Lacroix. Products of random matrices with ...

**0**

votes

**1**answer

249 views

### A particular kind of Cauchy Principal Value integral

I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form
$$\mathcal{P} ...

**2**

votes

**2**answers

787 views

### Singular Value Decomposition of Noisy Matrices

I am an engineer who makes measurements of a variable over a grid
of, say, $m\times n$. Since these are actual measurements, the true
values are always corrupted by noise, and what I measure is a ...

**1**

vote

**1**answer

142 views

### Scaling laws for singular values of random matrices

Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular ...

**2**

votes

**1**answer

187 views

### invertibility of a matrix with a Gaussian perturbation

Suppose that $A$ is an arbitrary fixed $n\times n$ matrix and $G$ a random $n\times n$ matrix with i.i.d. $N(0,1)$ entries. Is there a simple proof that $A+G$ is invertible with probability 1?
What ...

**1**

vote

**0**answers

64 views

### Random Schrödinger operators with asymmetric Lifshitz tails?

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...

**10**

votes

**1**answer

643 views

### Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...

**21**

votes

**0**answers

980 views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...

**2**

votes

**0**answers

108 views

### Quantifying the amount of structure in a data set via random matrix theory

Given a data matrix, $M \in \mathbb{R}^{n \times p}$, I am interested in methods quantifying the amount of structure in present in $M$.
I've found a few approaches, but I would like to learn more ...

**0**

votes

**1**answer

243 views

### Stochastic processes with random matrices

I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that ...

**7**

votes

**0**answers

229 views

### How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are $m$ independent $d\times d$ random matrices and let $\overline{X} = \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices ...

**15**

votes

**4**answers

486 views

### The latice spanned by $m$ random 0-1 vectors of length $n$

Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...

**1**

vote

**0**answers

306 views

### Monte Carlo sampling high dimensions with the halton sequence?

Referring to the Halton Sequence, Swiler et al 2006 state that
In cases where a large number of input variables are sampled,
Robinson and Atcitty recommend using a leaped sequence, where the
...

**3**

votes

**0**answers

194 views

### Concentration of functions of random unitary matrices

Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials ...

**9**

votes

**2**answers

874 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

455 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

392 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

299 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

208 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

280 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

147 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

119 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

387 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

446 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

431 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

469 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

115 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

840 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

395 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

324 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

919 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

405 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

522 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 ...

**16**

votes

**0**answers

467 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 ...

**4**

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

374 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

327 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

849 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

477 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 \\
...

**5**

votes

**4**answers

3k 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

300 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

445 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 ...