Statistics of spectral properties of matrix-valued random variables.

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3
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2answers
247 views

Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions: (...
1
vote
2answers
113 views

Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables. Are there cases where one can been able to quantify $...
3
votes
1answer
54 views

Matrix model for “$\beta$-Ginibre” ensembles

A very well known result in random matrix theory is that there exists "nice" (i.e., with independent entries) tridiagonal matrix for the $\beta$-ensembles of random matrix theory $$\propto\prod_{i<...
0
votes
0answers
35 views

Is that possible to use stieltjes transform for multiple matrices

I have the matrix calculation with expression \begin{equation} \frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1} \end{equation} whereas $\mathbf{H} \in \mathbb{C}^{M\times K}$, $\mathbf{...
4
votes
2answers
155 views

Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
2
votes
1answer
294 views

Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
1
vote
1answer
121 views

Spacing of the largest singular values of Wishart matrix

Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...
1
vote
1answer
73 views

Upper tail concentration of sample covariance matrices

I'm interested in concentration of the following random matrix sum in spectral norm $\frac{1}{m}\sum_{k=1}^m b_k^2\mathbf{a}_k\mathbf{a}_k^*$ Here $\mathbf{a}_k\in\mathbb{R}^n$ are i.i.d. standard ...
2
votes
0answers
124 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
0
votes
0answers
34 views

Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, $$\lvert\lambda_1-...
3
votes
1answer
101 views

Integrability of complex gaussian random matrix model

It is known that the partition function $$ \mathcal{Z}_1=\int dH e^{-N{\rm Tr}(H^2)}e^{-NV(H)},$$ where the integral is over $N\times N$ hermitian matrices $H$, with the potential $$ V(H)=\sum_{j\ge 1}...
25
votes
1answer
2k views

When should we expect Tracy-Widom?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
1
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0answers
31 views

On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set $$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$ We can further define the ...
4
votes
1answer
80 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 ...
6
votes
1answer
117 views

Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
1
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0answers
42 views

Majorizing inequality on spectral norm of product of a random and a deterministic low-rank projection

Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean ...
1
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0answers
42 views

Explanation for a spectral measure [closed]

Could someone help me, please, to understand in term of entries of a Matrix $M=(m_{i,j})_{i,j\in\{1,n\}^2}$ the following measure : $$ \frac1{n} \sum_{i=1}^n \langle v_i,e_j \rangle \delta_{\lambda_i}...
9
votes
3answers
851 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 $v$ ...
2
votes
0answers
142 views

Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem. Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = \...
2
votes
0answers
47 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
4
votes
0answers
105 views

Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$ I am ...
2
votes
1answer
119 views

Averages of vector inner products over the Haar measure

Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is? $$ \int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi $$ ...
3
votes
1answer
133 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 ...
1
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2answers
73 views

Non-asymptotic bound on the variance of largest singular value of gaussian matrix

Let $A$ be a gaussian matrix of size $d \times n$ where all the coefficients are drawn i.i.d. from $ \mathcal{N}(0, 1)$ and denote by $s_{\text{max}}$ its largest singular value. Theorem 2.6 of http:/...
-1
votes
2answers
179 views

Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
2
votes
0answers
114 views

Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection

Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...
4
votes
2answers
279 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, ...
17
votes
5answers
1k views

Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
1
vote
1answer
93 views

A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
1
vote
1answer
60 views

Normalizing Entries In Defining Random Matrices (Wigner Matrix)

In the definition of Wigner Matrix (a certain type of random Matrices) we take to independent family of i.i.d zero mean distributions $\{Z_{i,j}\}_{1<i<j}$ and $\{Y_{i}\}_{1\leq i}$ and then the ...
4
votes
0answers
54 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}} \...
3
votes
1answer
157 views

Bai and Silverstein's “Lemma on Quadratic Forms” - question about the constant $C_p$

In the book "Spectral Analysis of Large Dimensional Random Matrices" by Bai and Silverstein, there is the following lemma: Lemma B.26 (pg. 530) Let $A=(a_{ij})$ be an $n\times n$ nonrandom matrix ...
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0answers
24 views

Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately) $$ g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u)....
1
vote
1answer
91 views

If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space. Usually, we have a ...
2
votes
1answer
78 views

Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem: \begin{equation} \mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\...
8
votes
1answer
305 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
12
votes
1answer
311 views

Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices

If $A$ is chosen uniformly at random over all possible $n$ by $n$ Toeplitz (or circulant) (0,1)-matrices, can we give any bounds for the expected size of the determinant of $AA^T$? All arithmetic is ...
2
votes
1answer
134 views

A question from Zeitouni's Introduction to Random Matrices

I have a question regarding exercise 2.1.5 on page 19 in this book: http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf I would like a reference or help on this exercise. The exercise asks the ...
4
votes
0answers
115 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 ...
13
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0answers
349 views

How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
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votes
1answer
70 views

Alternative formula of a Green's function for average density of eigenvalues of random matrix

A Green's function is defined as follows: $$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$, where $I$ is the $N$-dimensional identity and $E$ means expectation value ...
2
votes
2answers
365 views

Distribution of dot product of two unit random vectors

Consider $\mathbf{u}, \mathbf{v}\in \mathcal{C}^M$ to be two independent unit norm random vectors on the $M-1$ dimensional complex sphere $\mathcal{S}^{M-1}$. In addition, $\mathbf{u}$ follows an ...
1
vote
1answer
85 views

Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix from Gaussian measure

I am now reading a paper by Sommers, H. J., et al. "Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898., it claims a mathematical statement (equation (2) in ...
8
votes
1answer
423 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
2
votes
1answer
130 views

A slight generalization of Mehta's integral.

I am trying to find the value of following integral $$\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\prod_{i=1}^ne^{-\frac{t_i^2}{2}+\alpha_i t_i}\prod_{1\le i<j\le n}\left|t_i-t_j\right|^{2\...
0
votes
1answer
84 views

Almost sure convergence of smallest eigenvector of diagonal matrix

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...
6
votes
1answer
160 views

Extension of Wigner's semicircle law?

It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries. My question is: is there any study about the more general ...
4
votes
1answer
206 views

Tail bounds on eigenvalue gaps for GUE

What I'm looking for is a non-asymptotic bound on the probability that the smallest gap between eigenvalues of a GUE matrix does not exceed a certain value. I'm aware of the bounds in http://imrn....
3
votes
1answer
109 views

Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. Amer....
1
vote
1answer
192 views

Sum of Eigenvectors Entries of an Adjacency Matrix

I have a question regarding the sums $\sum_{i=1}^{n}v_{j}\left(i\right)$ where $v_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length. Ordering the eigenvectors by ...