Statistics of spectral properties of matrix-valued random variables.

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220 views
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A conjecture about the entropy of matrix vector products

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
2
votes
1answer
65 views

limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...
0
votes
0answers
28 views

Derivative of Cholesky decomposition [migrated]

I would like to compute the derivative of the Cholesky decomposition, for example I have a matrix 2 x 2 R = 1 rho rho 1 where rho is a parameter, now I compute the Cholesky decomposition of ...
1
vote
0answers
16 views

the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...
3
votes
0answers
114 views

Eigenvalue Gap Probability Through Method of Moments

Let $M_n$ be drawn from $n\times n$ matrices under the Circular Orthogonal Ensemble (COE) distribution. Then the eigenvalues of $M_n$ all lie on the unit circle. Starting on the real line and going ...
1
vote
0answers
55 views

Distribution of the Gram Matrices

Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and ...
3
votes
2answers
77 views

Random matrices: why to distingusish bulk and edge cases?

I have been recently reading some papers on universality of spectral statistics of random matrices written by Terry Tao, Van Vu, L. Erdos, H.T. Yau and others, and I am puzzled by such a dichotomy, ...
2
votes
1answer
94 views

Distribution of the Gram matrix

Let $\mathbf{X}$ be an $m\times k$ random matrix ($m>k$) of rank $k$, having the density function $f_\mathbf{X}(X)$. What is the distribution of $\mathbf{Y}=\mathbf{XX}^T$? Basically my question is ...
2
votes
1answer
207 views

Probability distribution of uAv…

Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix. What is the distribution of $u^HAv$ ( or $||u^HAv||^2$) where : u is a column vector of U. v ...
1
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0answers
33 views

Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...
5
votes
1answer
169 views

Analysis of $AB^{-1}$, where $A,B$ are random matrices

I am looking for help pointing me in the direction of any literature or other known work that analyze the probability distribution or other important properties of random variables of the form ...
0
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0answers
102 views

A matrix rank problem over finite fields: Is that a known problem?

I have already asked the same question on cstheory.SE, but I haven't got an acceptable answer. So, I decided to ask it here. It might be a known problem, however. Let $A \odot B$ denote elementwise ...
0
votes
1answer
152 views

maximum of certain Gaussian processes

Let $\mathbf{a}_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the ...
2
votes
1answer
45 views

Spectral norm tail bound of a correlated random matrix

I am looking for the tail bound of spectral norm for certain type of random matrix. Let's say we have a $n\times n$ symmetric random matrix $R$, and for each entry $R_{ij}$, we have that $$ ...
7
votes
1answer
130 views

Distribution of entries of a doubly-sorted random matrix

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...
7
votes
1answer
178 views

Concentration of sum of powers of normals

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...
18
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1answer
1k views

Unexpected $\sqrt{3}$

A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as ...
2
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0answers
90 views

Finite Volume 1D Anderson Tight Binding Model

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the ...
2
votes
1answer
65 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
3
votes
1answer
56 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 ...
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0answers
46 views

Compare full-rank probabilities of products of random matrices

Consider two matrices $C_1=A\times B_1$ and $C_2=A\times B_2$, where $A\in\mathbb{F}_q^{N\times K}$, $B_1\in\mathbb{F}_q^{K\times M}$ and $B_2\in\mathbb{F}_2^{K\times M}$; $M\leq N\leq K$. It is ...
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0answers
43 views

integrality of a linear program — binary equality constaints

Consider the following linear program: $\left\{ \begin{array}{l} \underset{x}{max} \;\;c^Tx\\ [I, \;B]x = \mathbf{1}\\ x\geq 0 \end{array} \right.$ where $c$ is a vector ...
4
votes
1answer
123 views

Estimating the probability that $\|Av\| \ge \|v\|$

Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such ...
2
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0answers
49 views

Random square submatrices of a Hadamard matrix

Question: For $N$ be a power of $2$, let $A$ be a random $d \times d$ submatrix of the $N \times N$ Hadamard matrix (the matrix of the Hadamard/Walsh-Fourier transform). What is the best known upper ...
3
votes
1answer
70 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, ...
3
votes
1answer
109 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 ...
2
votes
1answer
137 views

Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
10
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0answers
213 views

What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...
1
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1answer
56 views

ordinary least square and random projection

Let $X$ a $d \times T$ given matrix and $M$ a $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$ where $'$ denotes the transpose ...
0
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1answer
62 views

Determine the expected size of a lower triangular sub-matrix of a random matrix?

Consider a $N\times N$ random matrix $A=[A_{ij}]$, whose elements are independently randomly chosen from the binary field $\mathbb{F}_2=\{0,1\}$ with probabilities $p_0=p$ and $p_1=1-p$. Suppose that ...
3
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0answers
122 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 ...
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1answer
144 views

Distribution of sum of freely independent Marchenko-Pastur measures

Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables ...
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1answer
202 views

Random matrix determinant problem

Suppose we have a a set of random matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are random complex valued ...
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0answers
466 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 ...
0
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1answer
175 views

Bounds on the eigenvalues of a random binary matrix

Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a_{i,j}=0)=P(a_{i,j}=1)=0.5$ (although I appreciate any advice on the case of ...
0
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1answer
229 views

Null space of random $(0,1)$ binary matrix [closed]

What can be said about the null space of random $(0,1)$ rectangular binary matrices? In particular, I am interested in the probability that there is any non-zero vector with only integer coordinates ...
0
votes
0answers
65 views

Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
2
votes
0answers
112 views

Distributions of eigenvalues for matrix normal distribution: related references

I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution. I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...
0
votes
1answer
112 views

Relation between the eigenvalue density and the resolvent?

Disclaimer: This is a cross-post from Math Underflow. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on this ...
5
votes
1answer
168 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 ...
2
votes
2answers
143 views

Probability for a random positive-semidefinite matrix to not be positive-definite?

If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite ...
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votes
1answer
207 views

Rank of a random matrix

Let $x$ a random Gaussian vector of size $n$ with i.i.d coefficients $N(0,1)$. Let $J$ a random matrix with i.i.d coefficients $N(0,\sigma^2/n)$ where $\sigma \in [0,1]$. For any integer T>n, define: ...
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0answers
109 views

expected inverse of circulant plus random diagonal

I have a deterministic circulant matrix $R$ and a random diagonal matrix $X$ where all elements are IID and positive. I need to determine the expected inverse of $R+X$, that is: Evaluate, in closed, ...
2
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0answers
56 views

Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices

I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...
3
votes
2answers
168 views

Eigenvalue distribution of the sum of two random matrices

Suppose $D$ is a diagonal matrix of size $n \times n$ with diagonal elements $D_{ii}$ which are independent standard centered Gaussian random variables. Then consider a matrix $J$ such that its ...
0
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0answers
224 views

expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$. Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
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0answers
50 views

Determining the asymptotic behavior of some scalar function of random matrices

Consider a series of random matrices $X_n\in\mathbb{R}^{n\times m}$ consisting of i.i.d. entries, each with zero mean and variance $1/m$, and let $y_n\in\mathbb{R}^{n\times1}$ be a random vector with ...
0
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1answer
116 views

concentration of sums of fourth moment of normals

I was wondering what is the best tail bound for \begin{equation*} \mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ? \end{equation*} where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.
3
votes
1answer
130 views

Condition number of a random 0-1 matrix

Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined ...
15
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0answers
261 views

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 ...