# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

62 questions
Filter by
Sorted by
Tagged with
4k views

6k views

### Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
4k 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 ...
4k 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 ...
5k views

### Anti-concentration bound for permanents of Gaussian matrices?

In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
2k views

### Expected value of determinant of simple infinite random matrix

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$ I would like to ...
1k views

### Determinant of real Wishart matrix

Suppose $A$ is a real $N \times P$ matrix, $P \geq N$, with entries drawn independently according to $A_{ij} \sim \mathcal{N}(0,1)$. Then $W = A \, A^\top$ is a member of the real Wishart ensemble. ...
514 views

### Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
584 views

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

If $A$ is chosen uniformly at random over all possible $m \times n$ (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 ...
2k views

### Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
2k views

### PT Symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
2k views

### Jensen Polynomials for the Riemann Zeta Function

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) the abstract includes In the case of the Riemann zeta function, this proves the ...
898 views

### How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
927 views

### Relationship between R-transform and free convolution of random matrices?

I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
2k views

### What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
369 views

### Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...
3k views

### Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix

TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...
223 views

### Riemann theta function inequality for a class of large random matrices

The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
256 views

### Phase transition in matrix

Playing around with Matlab I noticed something very peculiar: Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by $$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$ ...
617 views

### Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
288 views

### What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$?

Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too ...
2k views

2k 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 ...
603 views

543 views

### Determinant of a random row stochastic matrix

Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit ...
456 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 ...
197 views

### Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices

I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
501 views

### Upper bounds on the condition number of the eigenvector matrix

Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$. Question: Are there any upper bounds on the condition number of the ...
73 views

281 views

### Expectation of the trace of inverse of a Gaussian random matrix

Given a $N×M$ random complex gaussian matrix $X$ and $N×K$ random complex gaussian matrix $Y$ I'm interested in approximating the expectation expressed as: \begin{align} E[trace({(aX{X^H} + I)^{ - ...
124 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 ...
210 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 ...
213 views

### Is the inequality of the random matrices correct?

I am not familiar with random matrices but I need to confirm the correctness of the inequality below. Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let $A_1,\ldots, A_n$ be $m\times m$ ...
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 ...