Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
307 questions with no upvoted or accepted answers
36
votes
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answers
2k
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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 ...
- 188k
21
votes
0
answers
2k
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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 ...
- 24.7k
19
votes
0
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What does a product of many Gaussian matrices converge to?
Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$.
Is ...
- 2,442
18
votes
0
answers
469
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Quasi-classical limit of representation theory
I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
- 1,765
15
votes
0
answers
2k
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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
- 11.1k
14
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What is the reason the eigenvalues of GUE and CUE matrices tend locally to the same distribution?
It's well known in random matrix theory that locally the eigenvalues of a random matrix from the Gaussian unitary ensemble tend to a sine-kernel determinantal point process. Likewise, locally the ...
- 2,151
13
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0
answers
809
views
Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it
In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
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13
votes
0
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591
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What are the difficulties in proving almost-everywhere stability of Gaussian elimination?
It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
12
votes
1
answer
628
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A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
- 1,608
11
votes
0
answers
366
views
Lower Bound on the Volume of Certain Polytopes
Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks
$$
\rho=\{B_1,B_2,\ldots,B_{k}\}
$$
we can define the set of equations
$$
E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in B_{i}}{x_j}\quad\text{...
- 3,626
9
votes
0
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454
views
Framework for primes vs random matrices
This is inspired by What results would follow from or imply "randomness" of the primes? , but I think it is sufficiently different to ask separately.
We can formalise probability in ...
- 56.9k
8
votes
0
answers
170
views
Random walk on matrix until singularity
Consider a random walk on matrices, where one starts with the matrix $M=I_n$ and at each step randomly chooses an entry of $M$ to increase by $1$.
I’m interested in two things about this walk:
What’s ...
- 541
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,442
8
votes
0
answers
318
views
Maximum probability of a set of vectors from $ \mathbb{F}_2^n $ being linearly independent
Suppose $ m $ vectors from the vector space $ \mathbb{F}_2^n $ are selected independently according to a distribution $ P $ over $ \mathbb{F}_2^n $. Here $ \mathbb{F}_2 $ denotes the field with two ...
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8
votes
2
answers
547
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Concentration inequality for minimal eigenvalue of sample covariance
I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...
- 323
8
votes
0
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183
views
Can the GUE be thought of as a uniform point in a high-dimensional polytope
I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
- 22.8k
8
votes
0
answers
348
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 - zI)^\...
7
votes
0
answers
224
views
Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$
We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
- 745
7
votes
0
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758
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Product of two random Gaussian matrices - orthant probability
Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
- 968
7
votes
0
answers
179
views
Can one "smooth over" k-wise independence to get actual independence?
I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...
- 91
7
votes
0
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216
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irregular LDPC code construction algorithm
I want to construct a sparse random binary matrix ${{\bf{H}}_{m \times n}}$ that has the following properties
1- Faction of columns of weight $i$ is ${v_i}$ .
2- Fraction of rows of weight $i$...
- 275
7
votes
0
answers
396
views
Stable distributions for Lindeberg exchange strategy?
Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
- 2,350
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
0
answers
244
views
$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now.
In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
- 3,691
5
votes
0
answers
161
views
Determinant bounded away from zero
Suppose $P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1})$ is a noncommutative polynomial with positive coefficients. We may then consider the map $g:U(n)^k\rightarrow\mathbb{C}$ from the unitary group $U(n)$,...
- 163
5
votes
0
answers
239
views
Expected value of $X^{\top}(XAX^{\top})^{-1}X$ for large random $X$
Let $X\in \mathbb{R}^{m\times n}$ be a random matrix where the entries are i.i.d. standard normal, and let $A\in \mathbb{R}^{n\times n}$ be a deterministic diagonal matrix with positive entries on the ...
- 161
5
votes
0
answers
163
views
On $\ell_1$ to $\ell_1$ operator norm of matrix with inverse Wishart distribution
Consider a random $n\times p$ matrix $X$ with $n\ll p$ and all entries of $X$ i.i.d. standard normal. For this $X$, the system of linear equations $y=Xw$ has infinitely many solutions, and the one ...
- 151
5
votes
0
answers
235
views
Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
- 686
5
votes
0
answers
121
views
Spectrum of sum of fixed matrices with random signs
Let $A_1,\ldots,A_k$ be a given sequence of $N$-by-$N$ Hermitian matrices. Assume all have spectrum contained in $[-1,-\delta] \cup [+\delta,+1]$ for some $\delta>0$. Let $$A=\frac{1}{\sqrt{k}} \...
- 547
5
votes
0
answers
133
views
Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$
Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
- 1,997
5
votes
0
answers
327
views
Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
- 161
5
votes
0
answers
221
views
Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
5
votes
0
answers
396
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 $p(U,V)$...
- 2,449
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
4
votes
1
answer
332
views
Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
- 6,853
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
4
votes
0
answers
187
views
Dyadic distribution of $0/1$ permanents
Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.
What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents
in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
- 13.9k
4
votes
0
answers
102
views
What do the eigenvalues of a random element of $\mathbb Z_\ell[\Gamma]$ look like?
Let $\Gamma = \varprojlim \Gamma_n$ be a profinite group with $\Gamma_n$ finite quotients. For concreteness, let us fix $\Gamma_n = \operatorname{PGL}_2(\mathbb Z/\ell^n)$ so $\Gamma = \operatorname{...
- 7,746
4
votes
0
answers
134
views
What is known about the density of states for the Anderson Model?
The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(\mathbb{Z}^d)$)
$$
H_\omega = - \triangle + V(\omega)
$$
where $V(\omega) \mid x \rangle = \omega(x) \mid x \rangle$ ...
- 1,054
4
votes
0
answers
637
views
Comparison of concentrations of different $L^p$-norms of (sub) Gaussian distributions
It's well-known that the Euclidean $2$-norm of subgaussian random vectors concentrates in high dimensions, e.g. when $X \sim \mathcal{N}(0,I_n),$ (or in general $X$ is subgaussian with independent co-...
- 1,467
4
votes
0
answers
625
views
Random matrices from QR
In the very nice paper by GW Stewart:
Stewart, G. W., The efficient generation of random orthogonal matrices with an application to condition estimators. (With mircofiche section), SIAM J. Numer. ...
- 96.4k
4
votes
0
answers
35
views
Product of values of a matrix-valued function over $S^1$
Assume you have a function $f:S^1 \rightarrow \mathrm{GL}_d(\mathbb{C})$ whose coefficients are Laurent polynomials $f_{i,j}(q)\in \mathbb{C}[q^{\pm 1}].$
I am interested in getting conditions for ...
- 492