Tagged Questions

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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 ...
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$\epsilon$-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 ...
211 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 ...
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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 ...
189 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 ...
131 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.)
120 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 ...
358 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 ...
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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 ...
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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 ...
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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 ...
360 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 ...
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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 ...
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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 ...
740 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 ...
307 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 ...
742 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 ...
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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) ...
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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 ...
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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 ...
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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 ...
608 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 ...
386 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: ...
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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 ...
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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: $e^{-Q} = S$?
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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 ...
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, ...