0
votes
1answer
121 views

How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X

Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
4
votes
2answers
781 views

Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form $$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix} $$ where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
2
votes
2answers
615 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
0answers
56 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 $\sigma(H)$ ...
22
votes
0answers
865 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 ...
0
votes
1answer
391 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 ...
6
votes
1answer
563 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. ...
4
votes
2answers
560 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 ...
7
votes
0answers
510 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 ...
5
votes
2answers
489 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 ...