# Tagged Questions

**9**

votes

**1**answer

226 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 ...

**1**

vote

**0**answers

87 views

### $\epsilon$-covering number of a set of rank-2 matrices

Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...

**2**

votes

**2**answers

220 views

### Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation,
$$ E\left[ \exp(\mathbf{xx}^\top)\right]$$
where $\exp(\cdot)$ is element-wise exponential function (not ...

**8**

votes

**1**answer

265 views

### An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have
$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...

**6**

votes

**1**answer

271 views

### Injectivity of matrix “fingerprint”

Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ ...

**4**

votes

**1**answer

116 views

### Weak ergodicity of nonhomogenous products of 0-1 matrices

Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...

**0**

votes

**1**answer

201 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 ...

**2**

votes

**1**answer

132 views

### Number rank-k 0-1 matrices (characteristic 0)

What is the number of $n\times n$ 0/1-matrices with rank $k$?
(The rank is taken over the rationals.)

**2**

votes

**0**answers

124 views

### Random variable matrix exponential

I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here.
What ...

**10**

votes

**1**answer

359 views

### A simple proof for a theorem of Szekeres and Turán

Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...

**8**

votes

**0**answers

285 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

**20**

votes

**4**answers

1k views

### Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...

**1**

vote

**2**answers

169 views

### Strictly positive definite autocovariance function of fGn

Hi,
let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter.
I want to show that $\gamma$ is ...

**10**

votes

**1**answer

368 views

### Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...

**5**

votes

**1**answer

236 views

### Convergence rate for product of stochastic matrices

Hi,
I have a system of the form $$x(t+1) = A(t + 1) x(t),$$ for $t \geq 1$, and some fixed initial condition $x(1)$. Here $A (t)$ is a time-varying $m \times m$ matrix that is stochastic at all ...

**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 ...

**15**

votes

**0**answers

748 views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**1**

vote

**1**answer

319 views

### Lower bound on Bhattacharya distance between independent Gaussian distributions ?

I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions ...

**15**

votes

**5**answers

747 views

### What fraction of n x n invertible integer matrices contain at least one unit?

The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure ...

**4**

votes

**1**answer

153 views

### Set of unitaries with “spread-like” properties

I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that:
$
\sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} ...

**2**

votes

**1**answer

675 views

### Bochner's Theorem and Total Positivity

Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...

**3**

votes

**1**answer

227 views

### Estimating spectral radius with a Gaussian vector

Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$,
and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.
Is the following lemma true:
If the ...

**6**

votes

**4**answers

675 views

### Doubly stochastic matrices as squares of entires of unitary matrices

Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a ...

**1**

vote

**1**answer

357 views

### Random sampling a symmetric matrix

I'd like to sample the elements of a symmetric square matrix uniformly. For example, for a $N\times N$ matrix, I'd like to only keep $\alpha$% of the matrix elements to build a sparse matrix, while ...

**1**

vote

**0**answers

242 views

### The finite-dimensional distributions of infinte-dimensional limit of finite-dimensional vectors

Suppose we have the process $\{\varepsilon_t,t\in \mathbb{N}\}$. Suppose that this the finite-dimensional distributions of this process are Gaussian, i.e. for any $t_1,...,t_n$, vector ...

**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 ...

**2**

votes

**0**answers

175 views

### Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...

**13**

votes

**3**answers

1k views

### Distribution of the spectrum of large non-negative matrices

This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) ...

**8**

votes

**1**answer

504 views

### Bounds on $||P^{k+1} - P^k||$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k>>n$.

The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...

**1**

vote

**2**answers

451 views

### Markov chain convergence problem.

Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)
2- For ...

**-2**

votes

**3**answers

659 views

### Convergence of a markov matrix

Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)
2- Not all ...

**15**

votes

**3**answers

609 views

### Random products of projections: bounds on convergence rate?

The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...

**2**

votes

**1**answer

391 views

### Multinomial transformation for matrices

Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
...

**27**

votes

**3**answers

2k views

### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...

**3**

votes

**2**answers

1k views

### Matrices whose exponential is stochastic

The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?

**5**

votes

**1**answer

517 views

### Spectrum of a generic integral matrix.

My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit ...

**5**

votes

**1**answer

502 views

### Are there interesting problems involving arbitrarily long time series of small matrices?

Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...

**4**

votes

**4**answers

852 views

### Distribution of 1-norm for Gaussian Unitary Ensemble

Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this ...