# Tagged Questions

**2**

votes

**0**answers

34 views

### Random matrices whose limit gives exact Wigner surmise

Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write ...

**9**

votes

**1**answer

227 views

### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...

**0**

votes

**0**answers

64 views

### Bounding multiplications of PSD random matrices

Consider the following setup,
$(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$.
The ...

**1**

vote

**1**answer

102 views

### Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let
\begin{equation*}
p = p\left(\kappa,d\right) := ...

**1**

vote

**0**answers

20 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 ...

**1**

vote

**0**answers

34 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 ...

**0**

votes

**0**answers

50 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 ...

**3**

votes

**0**answers

140 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**

votes

**1**answer

223 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:
...

**0**

votes

**0**answers

242 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 ...

**3**

votes

**1**answer

139 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 ...

**0**

votes

**1**answer

212 views

### Determine the probability that two random vectors over a finite field are orthogonal

Hi all,
Suppose that $\mathbf{f}=[f_1, f_2,\ldots,f_m]$ and $\mathbf{g}=[g_1,g_2,\ldots,g_m]$ are two $m$-dimensional vectors. All $f_i$'s are chosen uniformly randomly from a finite field ...

**2**

votes

**0**answers

119 views

### Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate.
Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...

**9**

votes

**2**answers

611 views

### Probability of random (0,1) Toeplitz matrix being invertible

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...

**1**

vote

**1**answer

82 views

### Expected rank - computable approximations

I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general).
Computing $\mathbb{E} \ ...

**27**

votes

**2**answers

1k views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...

**0**

votes

**0**answers

97 views

### Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?

Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...

**9**

votes

**2**answers

194 views

### Iterating Random Matrix Operations

Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...

**1**

vote

**1**answer

142 views

### Scaling laws for singular values of random matrices

Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular ...

**21**

votes

**0**answers

949 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 ...

**7**

votes

**2**answers

443 views

### Maximum Singular Value of a random +1/-1 matrix

Hi,
Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability 0.5 to be either +1, or -1. Do you know any result in the ...

**1**

vote

**3**answers

405 views

### Eigenvalues of Krylov matrices

Let an $n\times n$ matrix ${\bf A}$, the all ones vector ${\bf w}$, and the $n\times n$ Krylov matrix
$${\bf K}_n = \left[ {\bf w}\;\;{\bf A}{\bf w}\;\;\ldots \;\; {\bf A}^{n-1}{\bf w}\right].$$
Is ...

**7**

votes

**3**answers

474 views

### Relationship between free probability and deterministic graphs?

Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
...

**8**

votes

**1**answer

836 views

### Matrix inversion lemma with pseudoinverses

The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.
Suppose we pick $n$ values ...

**12**

votes

**1**answer

757 views

### A Question on Random Matrices

Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by
$$
V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q})
$$
where $x_{1},x_{2},\ldots,x_{n}$ are iid ...

**3**

votes

**4**answers

530 views

### efficient way to compute the inversion of the following matrix

Hi, there
I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...

**15**

votes

**1**answer

1k views

### Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...

**5**

votes

**2**answers

499 views

### Dependence of trace norm on matrix size for smooth vs. random matrices.

Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...