Statistics of spectral properties of matrix-valued random variables.

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2
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1answer
687 views

Expectation of product of Gaussian random vectors

Say we have two multivariate Gaussian random vectors $p(x_1) = N(0,\Sigma_1), p(x_2) = N(0,\Sigma_2)$, is there a well known result for the expectation of their product $E[x_1x_2^T]$ (matrix result) ...
2
votes
1answer
494 views

What are the origin and applications of this result?

In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose $$ M = \begin{bmatrix} A & B \\\\ B^T & C ...
7
votes
0answers
519 views

Matrix integral identity

1) How to prove that $N\times N$ matrix integral over complex matrices $Z$ $$ \int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det ...
0
votes
1answer
500 views

randomized SVD singular values

randomized SVD decomposes a matrix by extracting the first k singular values/vectors using k+p random projections. my question concerns the singular values that are output from the algorithm. why ...
15
votes
1answer
1k views

Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following: Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
30
votes
1answer
3k views

Anti-concentration bound for permanents of Gaussian matrices?

In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of ...
5
votes
1answer
253 views

Expected inverse determinant with independent rows

Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors. More ...
4
votes
0answers
193 views

q-deformation of the unitary group integral

There is a well-known orthogonality property of $U(N)$ group characters $$ \int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu} $$ where the integral is ...
0
votes
2answers
247 views

Question about “wide” random matrices

Let $A \in \mathbb{R}^{m \times n}$ be a random matrix with i.i.d. entries (the distribution is not important), where $m < n$ (i.e. $A$ is a "wide" matrix). I would like a lower bound on $$ ...
5
votes
2answers
493 views

Dependence of trace norm on matrix size for smooth vs. random matrices.

Problem Consider two d x d complex matrices, R and S, whose entries lie in the unit disk: $\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $. Say that R is constructed by randomly choosing ...
2
votes
4answers
409 views

any known universality results of random matrices with non-independent entries?

GUE, GOE, hermitian/symmetric wigner matrices, ... already known with some mild assumptions. but is there any this kind of results without independent entries condition. thanks a lot.
11
votes
2answers
912 views

Evaluation of a combinatorial sum (that comes from random matrices)

I'm looking for an elementary combinatorial/generating function/etc proof of the following result: For nonnegative integers $r$, $$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} ...
12
votes
4answers
947 views

An experiment on random matrices

A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...
9
votes
1answer
646 views

U(3) Sato-Tate measure.

An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix ...
10
votes
3answers
960 views

Random Walks and Lyapunov exponents

Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $GL_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log||Y_1||)$ is finite, there exists a constant ...
6
votes
1answer
454 views

distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field

This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...
0
votes
1answer
211 views

Probability on the distance

Let $A$ be an $n\times n$ gaussian matrix whose entries are i.i.d. copies of a gaussian variable, and $\left\{ a_{j}\right\} _{j=1}^{n}$ be the column vectors of $A$. How to show that the probability ...
12
votes
7answers
4k views

Expected determinant of a random NxN matrix.

What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
8
votes
0answers
281 views

A formula for moments of the limit distribution of singular values in the proof of the circular law

One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix $(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - ...
6
votes
5answers
775 views

L-functions and random matrices

I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one ...
2
votes
4answers
2k views

Symmetrical Presentation of 4-Dimensional Rotation Matrix

This question is not urgent; just a matter of curiosity... It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...
8
votes
3answers
804 views

Singularity of sparse random matrices

The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...