Statistics of spectral properties of matrix-valued random variables.

**1**

vote

**1**answer

97 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 ...

**0**

votes

**0**answers

7 views

### An upper bound for the trace of the product of the (rank-deficient) sample covariance and its “regularized inverse”

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_p)$. Let $\hat S = \sum_{i=1}^n x_i x_i^T$ be the sample covariance.
When $p>n$, $\hat S$ is not invertible. However, by adding a positive-definite diagonal ...

**3**

votes

**0**answers

95 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 ...

**2**

votes

**1**answer

93 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 ...

**8**

votes

**6**answers

4k views

### Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it ...

**6**

votes

**1**answer

213 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**

votes

**1**answer

144 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 ...

**1**

vote

**1**answer

190 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

**1**answer

47 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 ...

**3**

votes

**0**answers

19 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 $\mathbb{R}^d$, ...

**-1**

votes

**1**answer

32 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 ...

**1**

vote

**1**answer

48 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

**1**answer

91 views

### Ordinary least square and random projection

Let $X$ be a given $d \times T$ matrix, and let $M$ be an $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$, where $'$ denotes the ...

**0**

votes

**0**answers

22 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 ...

**0**

votes

**0**answers

74 views

### On the least singular value of a random matrix and its minors

Let $A$ be an $n \times n$ random matrix with entries i.i.d from the continuous uniform distribution ${\rm U}([-1,1])$. Is it true that the least singular value of $A$ and all its minors is greater ...

**1**

vote

**0**answers

59 views

### Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...

**6**

votes

**1**answer

204 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} & ...

**6**

votes

**0**answers

70 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 ...

**2**

votes

**1**answer

179 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
...

**25**

votes

**0**answers

1k views

### 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 ...

**3**

votes

**0**answers

67 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 ...

**2**

votes

**0**answers

306 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

**0**answers

51 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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vote

**1**answer

308 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 ...

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vote

**0**answers

49 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 ...

**1**

vote

**0**answers

41 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 ...

**6**

votes

**1**answer

97 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 ...

**0**

votes

**2**answers

343 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 ...

**5**

votes

**1**answer

155 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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vote

**3**answers

123 views

### Extending GUE to a measure on operators?

Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$:
$$
\mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ ...

**6**

votes

**3**answers

273 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 ...

**0**

votes

**0**answers

37 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 $N(0,I_n)$ ...

**4**

votes

**0**answers

64 views

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

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

**1**

vote

**1**answer

130 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 ...

**11**

votes

**1**answer

373 views

### Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements
are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?
For example, for $n=3$ and $k=2$, the first ...

**3**

votes

**1**answer

53 views

### Vanishing Restricted Isometric Constant

In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are ...

**7**

votes

**7**answers

607 views

### Semicircle law universality elsewhere

Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...

**24**

votes

**0**answers

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**

vote

**0**answers

42 views

### Products of random permutations with fixed matrix

This question originates from an engineering problem, which I am solving. Any related references are highly appreciated.
Let $M_k(T)=\prod_{t=1}^T P_t S_k$ over some field (finite or reals), where ...

**2**

votes

**4**answers

898 views

### 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,

**0**

votes

**0**answers

151 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 ...

**0**

votes

**0**answers

33 views

### Stochastic independence of columns of projection matrix to the rest of the columns of a random matrix

First let me describe the setting of the problem.
I have a random matrix $A\in \mathbb{R}^{m\times n},\ (m<n)$ with $a_{ij}\sim \mathcal{N}(0,I)$ i.i.d. Let there be a given set of $K (K<m)$ ...

**18**

votes

**2**answers

756 views

### Laws of Iterated Logarithm for Random Matrices and Random Permutation

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then
$$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...

**0**

votes

**0**answers

71 views

### eigenvalue distribution of random projection

Suppose that $A$ is an $n\times n$ diagonal matrix with positive diagonal elements and $\Pi$ is a random $k\times n$ matrix that could be
(a) i.i.d. Gaussian, or
(b) $k$ rows of a random orthogonal ...

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vote

**0**answers

42 views

### Projection of a ray onto a random polytope

Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by ...

**8**

votes

**1**answer

329 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 ...

**1**

vote

**0**answers

59 views

### Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...

**4**

votes

**2**answers

322 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

**17**

votes

**2**answers

461 views

### Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...

**13**

votes

**1**answer

662 views

### Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...