Statistics of spectral properties of matrix-valued random variables.

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33 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 ...

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37 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 ...

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41 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 ...

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**2**answers

107 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 ...

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95 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 ...

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votes

**1**answer

110 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?
$$
\int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi
$$
...

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**1**answer

110 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 ...

**2**

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107 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 ...

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**2**answers

64 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 ...

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173 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 ...

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**1**answer

57 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 ...

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**0**answers

50 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}} ...

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22 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 + ...

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**1**answer

89 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**

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**1**answer

74 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](:= ...

**8**

votes

**1**answer

294 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 ...

**1**

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**1**answer

88 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, ...

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**1**answer

128 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 ...

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**0**answers

110 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 ...

**0**

votes

**1**answer

69 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 ...

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**1**answer

84 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 ...

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votes

**1**answer

83 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 ...

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**1**answer

149 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 ...

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votes

**1**answer

88 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 ...

**4**

votes

**1**answer

205 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
...

**3**

votes

**1**answer

101 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. ...

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**1**answer

169 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 ...

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**0**answers

132 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

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**1**answer

369 views

### How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with ...

**3**

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**1**answer

165 views

### Smallest eigenvalue gap of a non-symmetric random matrix

The question: Let $A$ be the matrix whose each element is an independently generated random variable which is uniform on $[0,1]$. One can see that the eigenvalues of $A$ will be distinct almost ...

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36 views

### Bound on principal angle of uniform random subspaces of different dimensions?

This paper derives the distribution of the largest principal angle between two subspaces sampled (independently) uniformly from the Grassmanian manifold of $p$-dimensional subspaces in $\mathbb{R}^d$, ...

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**1**answer

43 views

### finding a unitary submatrix inside a random matrix

Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...

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**1**answer

147 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:
...

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votes

**1**answer

151 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 ...

**0**

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**0**answers

36 views

### Order statistics of the diagonal terms of an inverse Wishart matrix

I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then ...

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**1**answer

215 views

### Is there a way to simplify the following trace expression?

I'd like to simplify the following expression:
$$\text{tr}\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}_{M\times M} & \mathbf{0}_{M\times N} \\ \mathbf{0}_{N\times M} & ...

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**0**answers

94 views

### 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 ...

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vote

**1**answer

95 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 ...

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**0**answers

109 views

### Upper bound on largest singular value of a heavy tailed random matrix

Let $A$ be a $k\times n, k<n-1$ random matrix with entries drawn i.i.d. from a standard Gaussian, and $B$ a $k\times m$ random matrix with entries drawn i.i.d from a standard Gaussian, and ...

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**0**answers

63 views

### $l_{\infty}$ norms of matrix perturbations

Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that ...

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**0**answers

332 views

### Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way.
All the ${\lambda}_i$ are distributed the same way with chi-square ...

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votes

**1**answer

125 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 ...

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**0**answers

54 views

### Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph.
Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...

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**1**answer

120 views

### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...

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**1**answer

157 views

### Modification of matching

Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...

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**0**answers

38 views

### schatten 1-norm of rank $k$ matrix

I am looking for a high-probability lower bound for the following rank-$k$ matrix
$$
X = u_1 v_1^T + u_2 v_2^T + \cdots + u_k v_k^T,
$$
where $u_1,\dots,u_k,v_1,\dots,v_k$ are independent $N(0,I_n)$ ...

**0**

votes

**2**answers

353 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

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votes

**3**answers

299 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 ...

**12**

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**1**answer

310 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 ...

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**1**answer

185 views

### Expected size of determinant of $AA^T$ for non-square random $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all ...