Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
871 questions
6
votes
2
answers
1k
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 this,...
- 96.4k
6
votes
1
answer
894
views
Expected value of orthogonal projection $X^{+}X$
Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\...
- 161
6
votes
3
answers
425
views
Probability a random matrix contains a short integer vector in its kernel
Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
- 3,377
6
votes
1
answer
353
views
Quaternion Wishart matrices of half-integer dimension?
For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
$$P_\beta(\...
- 188k
6
votes
2
answers
2k
views
Marginal distribution of the diagonal of an inverse Wishart distributed matrix
This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here:
Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
- 269
6
votes
1
answer
283
views
L^1-norm of the trace on the unitary group
Is the value of the following integral over the unitary group with respect to the normalized Haar measure known?
$$
\int |Tr(U)|^pdU.
$$
There are some results for $p=2k$, $k\in\mathbb N$ (see ...
- 103
6
votes
1
answer
346
views
Do random asymmetric games have more complicated strategies than random symmetric games?
Let $\Delta \subset \mathbb R^n$ be the locus of vectors whose entries are nonnegative and sum to $1$.
For $M$ an $n\times n$ matrix over $\mathbb R$, let $x_M \in \Delta$ be the vector $x$ that ...
- 148k
6
votes
1
answer
537
views
"Ergodicity" for eigenvalues of random matrices?
Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout).
Is there some "ergodicity-type" result for Wigner's semicircle ...
- 902
6
votes
0
answers
203
views
Spectrum of $\prod_i^d \left(I-x_ix_i^T\right)$ for isotropic $x_i$
Suppose $x_i\in \mathbb{R}^d$ are IID isotropic random vectors with $\|x_i\|=1$ and matrix $A_d$ is defined as follows:
$$A_d=\prod_i^d \left(I-x_ix_i^T\right)$$
Is anything known about the spectrum ...
- 2,442
6
votes
0
answers
279
views
Estimating $E[\operatorname{Tr}(ABABBA..)]$ for random shuffling of $A,B$?
How can I estimate the following value where $A,B$ are $d\times d$ matrices and expectation is taken over all random permutations of the product?
$$E_\text{shuffle}[\operatorname{Tr}\underbrace{AA\...
- 2,442
6
votes
0
answers
396
views
Typical eigenspectrum of a random projection of a large matrix
Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection
$$ A =...
- 453
6
votes
0
answers
295
views
Dimension-free sample complexity for estimating Gaussian covariance
(also asked on math.se, with no answers)
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$$...
- 2,442
6
votes
0
answers
554
views
a variation on Hanson-Wright inequality
The classic Hanson-Wright inequality states that for a Gaussian random vector $\mathbf{x}\in\mathbb{R}^n$ distributed as $\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
- 859
6
votes
0
answers
375
views
Kasteleyn, Gessel-Viennot and eigenvalues
The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...
- 1,343
6
votes
0
answers
302
views
Behavior of eigenspaces of adjacency matrices of random graphs (not via perturbation theory)
For the sake of discussion, let us say that we have the adjacency matrix $A$ of a graph, on $n$ nodes, from a stochastic block model with 2 blocks. Another name for this (usually used in computer ...
- 1,731
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
6
votes
0
answers
1k
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 ...
- 1,890
5
votes
2
answers
382
views
What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$?
Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too ...
- 297
5
votes
1
answer
403
views
A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...
- 2,246
5
votes
4
answers
854
views
Expected value of the spectral radius of a random nonnegative matrix
I have two questions, the second of which is related to this question posed by Denis Serre.
Let $X$ be a random variable and suppose that $Y=|X|$ (e.g., $Y$ could be the folded normal distribution). ...
- 1,719
5
votes
1
answer
515
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 ...
- 2,449
5
votes
1
answer
3k
views
Eigenvalues and eigenvectors of Gaussian random matrices
Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues?
...
- 899
5
votes
1
answer
680
views
What is the Essential Difference Between Random Matrices and Random Graphs?
I have the impression, that random graphs and random matrices seem to be perceived and treated as separate areas of interest; I'm not an expert in either of the subjects, so maybe my impression is ...
- 13.2k
5
votes
1
answer
400
views
Lower tail of random rank one sums?
Let $\{x_i\}_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$;
assume that $\|x_i\|_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x_1 \otimes x_1]$.
Define
$$
\Sigma_n =...
- 410
5
votes
2
answers
687
views
Dependence of trace norm on matrix size for smooth vs. random matrices.
Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...
- 846
5
votes
1
answer
2k
views
Distribution of eigenvalues of a Wishart matrix
Is there a known expression for the eigenvalue distribution of a matrix of the form
$$\sum_{i=1}^n k_ia_ia_i^T$$
where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)$ and $...
- 151
5
votes
2
answers
283
views
LDPC codes construction
According to Google Scholar original Gallager's article Low-density parity-check codes is cited more than 10000 times. It looks scary for non-experts.
I suspect that the number of algorithms for ...
- 12.3k
5
votes
1
answer
349
views
On the eigenvalues' distribution of random unitary
Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group.
For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...
- 1,503
5
votes
1
answer
533
views
Riemann-Hilbert approach to Selberg integral
I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
- 2,552
5
votes
1
answer
1k
views
A general formula for Gaussian integrals over matrix elements
The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I_\tau=\int \prod_{i, j=1}^{N} d J_{i ...
- 117
5
votes
1
answer
2k
views
Generating a random special unitary matrix
I could find many resources on generating random unitary matrices, usually citing F. Mezzadri, Notices of the AMS 54 (2007), 592-604 for a method which generates unitaries random with respect to the ...
- 163
5
votes
1
answer
2k
views
Relation between the eigenvalue density and the resolvent?
Disclaimer: This is a cross-post from math.stackexchange. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on ...
- 175
5
votes
2
answers
324
views
Is there a 'natural' projection from $O(n)$ into $S_n$?
Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties?
$F(P_\sigma) = \sigma$ for all $\sigma \in S_n$
$F^{...
- 1,295
5
votes
1
answer
759
views
How to draw a random normal matrix?
I would like to pick a random real normal (i.e. commuting with its transpose) matrix and I wonder if it can be done easily. I thought whether it would be possible to use a similar trick to drawing a ...
- 53
5
votes
1
answer
282
views
What is the spectral norm of a random projection times a diagonal?
Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix.
What can be said about the distribution of the largest ...
- 7,633
5
votes
1
answer
321
views
Random matrices having all real eigenvalues: uniform vs gaussian distributions
Let $P_n$ be the probability that a $n \times n$ real random matrix with independent entries and uniformly distributed on $[0,1]$ has all real eigenvalues.
Let $Q_n$ be the same probability, for a ...
- 318
5
votes
2
answers
294
views
Estimate on lowest eigenvalue in GOE
I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that ...
- 167
5
votes
1
answer
331
views
Matrix concentration bound
Suppose we have $N$ constant matrices $A_i \in R^{m\times m}, 1\leq i \leq N$. Consider $N$ random rotation-matrices $R_i \in SO(m), 1\leq i \leq N$. Is it possible to obtain a concentration bound on
$...
- 51
5
votes
2
answers
718
views
Is there any theoretical results about the determinants of a Non-Central Wishart matrix?
As we know that a Non-Central Wishart matrix is defined as
$W:=XX^T$, where $X \in \mathbb{R}^{p \times N}$, and
$X:= M + E$, with $M \in \mathbb{R}^{p \times N}$ a deterministic and non-zero matrix, ...
- 131
5
votes
1
answer
765
views
Measure concentration for weakly dependent random variables
For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-...
- 23k
5
votes
3
answers
4k
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 $...
- 383
5
votes
1
answer
312
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 ...
- 1,533
5
votes
1
answer
368
views
$(AB)^+\approx B^+A^+$ for $B$ "fat" enough?
Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse.
Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
- 2,712
5
votes
1
answer
473
views
Statistical independence of eigenvectors of real symmetric Gaussian random matrices
What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not ...
- 75
5
votes
1
answer
324
views
Symmetry type of non-cohomological automorphic forms
By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-...
- 809
5
votes
3
answers
1k
views
A conjecture about the entropy of matrix vector products
Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...
- 3,377
5
votes
1
answer
291
views
Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix
Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...
5
votes
1
answer
287
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 ...
- 4,466
5
votes
1
answer
284
views
concentration of random matrices involving normal random variables
Define the random variable
\begin{align*}
A=|a_1|^2\mathbf{a}\mathbf{a}^*
\end{align*}
where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as $\mathcal{N}(0,\mathbf{I}/2)+i\mathcal{N}(0,\...
- 859
5
votes
1
answer
694
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 ...