Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
871 questions
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Matrix integral identity
1) How to prove that $N\times N$ matrix integral over complex matrices $Z$
$$
\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Z)\det(1-x_2e^{AZ^\dagger}...
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randomized SVD singular values
randomized SVD decomposes a matrix by extracting the first k singular values/vectors using k+p random projections.
my question concerns the singular values that are output from the algorithm. why ...
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Smallest eigenvalue of a tricky random matrix
While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
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45
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Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
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Expected inverse determinant with independent rows
Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.
More ...
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4
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0
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q-deformation of the unitary group integral
There is a well-known orthogonality property of $U(N)$ group characters
$$
\int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu}
$$
where the integral is ...
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On the spectrum of random regular graph
For a random $d$-regular graph, where $d$ can be fixed or can grow slowly with the size of the graph $n$, what can we say about its spectrum - Do you believe it has simple spectrum?
Thank you,
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Question about "wide" random matrices
Let $A \in \mathbb{R}^{m \times n}$ be a random matrix with i.i.d. entries (the distribution is not important), where $m < n$ (i.e. $A$ is a "wide" matrix). I would like a lower bound on
$$
\phi(A)...
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5
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2
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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 ...
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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.
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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} \frac{1}{(p_0!)^2(p_1!)^2\...
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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 ...
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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 ...
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Random walks and Lyapunov exponents
Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $\mathrm{GL}_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y_1\|)$ is finite, there exists a ...
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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 ...
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1
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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
...
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7
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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?
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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 - zI)^\...
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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 ...
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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 ...
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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 ...
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