# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

446 questions

**2**

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67 views

### A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....

**3**

votes

**1**answer

72 views

### Spectral density of $D + XX^T$

Let $D$ be a fixed diagonal matrix with real entries, and $X$ a random $m\times n$ matrix. More precisely, the entries $X_{ij}$ are real independent and identically distributed. It can be shown that ...

**1**

vote

**1**answer

72 views

+50

### Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements

Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:
$\...

**0**

votes

**1**answer

29 views

### Probability of a quantity from Ginibre ensemble

I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have
$$P(s)=C_{N,K}\int.....

**2**

votes

**0**answers

63 views

### Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$

I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball:
If you pick n points uniformly at random from the surface of a d
dimensional sphere of ...

**1**

vote

**0**answers

92 views

### Error correcting codes via random matrices: How close to the Shannon bound?

I have a vague and probably rather naive question on error correcting codes.
Suppose we want to encode binary vectors of length $k$ as binary vectors of length $n$ in such a way that differences of ...

**3**

votes

**1**answer

100 views

### Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...

**2**

votes

**1**answer

120 views

### The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...

**1**

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**0**answers

29 views

### Expected value of inverse of complex non-central Wishart matrix

I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...

**1**

vote

**1**answer

81 views

### Can eigenvectors determine their original matrices given the basis of matrices space is small?

Let $A_1$, $A_2$, and $A_3$ be three different mutually orthogonal random hermitian operators, say dimension of $30\times 30$. For three random real numbers $a_1$, $a_2$, $a_3$, we have
$$
(a_1A_1+...

**2**

votes

**1**answer

80 views

### stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...

**2**

votes

**1**answer

80 views

### Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...

**1**

vote

**1**answer

55 views

### On the eigenvalue of the expectation value of a random matrix in quadratic form

When we handle with some dynamic input-output mappings, there occurs a question as follows:
Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...

**1**

vote

**1**answer

41 views

### Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?

Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:
$\...

**3**

votes

**1**answer

117 views

### Tail probability of random projection

Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...

**4**

votes

**1**answer

183 views

### integral kernel function for the SU(N) group

It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 ...

**-2**

votes

**1**answer

93 views

### On the Cauchy-Schwarz Inequality for trace function of random matrices

In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that
$$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$
which is the trace form of Cauchy-Schwarz-Inequality (CSI)....

**3**

votes

**0**answers

79 views

### Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate
$$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$
where the $\rho$ functions ...

**2**

votes

**1**answer

114 views

### What is the Essential Difference Between Random Matrices and Random Graphs?

I have the impression, that random graphs and random matrices seem to be perceived and treated as separate areas of interest; I'm not an expert in either of the subjects, so maybe my impression is ...

**2**

votes

**0**answers

56 views

### Recovering a rank-one matrix from its eigendecomposition after randomized rounding

Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting
$$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...

**0**

votes

**0**answers

140 views

### What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?

I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...

**1**

vote

**0**answers

32 views

### Singular values of random matrices with inhomogeneous variances

If $X$ is a random rectangular matrix with independent identically distributed entries of zero mean and equal variance, then as $X$ gets big its singular values tend to a Marchenko-Pastur distribution....

**2**

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**0**answers

74 views

### Matrix Chernoff sampling with out replacement

I am interested to know if the matrix Chernoff bound (see Theorem 5.1.1 in https://arxiv.org/pdf/1501.01571.pdf) holds if one samples without replacement.
For example, the Bernstein inequality is ...

**1**

vote

**0**answers

77 views

### How does the graph of percolation probability $\Pi$ vs. $p$ vary for different finite values of $L$?

This is a sequel to my previous question. @Carlo's response here (to my comment) prompted me to ask this question:
As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by ...

**0**

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**0**answers

56 views

### How to mathematically justify the “sampling” over only $100$ random matrices to estimate percolation thresholds?

As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by Stauffer et al., the variation of spanning cluster percolation probability $\Pi$ in a finite $L < \infty$ square ...

**1**

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**0**answers

139 views

### Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...

**0**

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**0**answers

38 views

### Dominating powers of a random matrix

Let $A_n$ be a (sequence of) random matrix such that $ A_n = (a_{ij})_{1 \leq i,j \leq n}$ and the $a_{ij}$ are iid, $\mathbb E\left[ a_{ij}\right] = 0 \quad E\left[ \vert a_{ij}\vert^2\right] = 1$.
...

**0**

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**0**answers

21 views

### Why do middle roots of the $\chi(p)$ graphs and percolation thresholds vary linearly with diagonal probability $q$ (in large random binary matrices)?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...

**2**

votes

**1**answer

71 views

### Generalization: (The “number” of) smaller sized clusters in large random binary matrices follow a descending order. Why?

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...

**3**

votes

**1**answer

134 views

### Why is number of single cell clusters always greatest in a random matrix?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...

**1**

vote

**0**answers

25 views

### What is the distribution of engenvalues of covariance matrix when the covariance has some block diagonal structure

Let's say we have a matrix $X \in \mathbb R^{n\times p}$, where $X_{i,j}$ sampled from a Gaussian $N(\mu, \sigma^2)$, we use $\Phi$ to denote $\{\mu,\sigma\}$ for simplicity.
Now, we sample $m$ ...

**3**

votes

**1**answer

220 views

### Approximating the expectation of a matrix inverse

Let
$$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$
where $A$ is a given $n \times m$ matrix (where $m \gg n$),
$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$
...

**3**

votes

**3**answers

151 views

### Random complex eigenvalues and averages of traces

I have asked this in MSE here, but got no interesting answers.
Suppose I have a random matrix $M$ of dimension $N$ which is real, but not symmetric. Suppose I know that, for large $N$, the marginal ...

**10**

votes

**0**answers

430 views

### multi-dimensional integral of modified Vandermonde determinant

I'm looking for suggestions on how one might try to compute the following $(N-1)$-dimensional integral:
$$I_N= \frac{1}{(2\pi)^{N-1}(N-1)!} \int\cdots\int \\
\begin{vmatrix}
1 ...

**3**

votes

**1**answer

89 views

### Statistical independence of eigenvectors of real symmetric Gaussian random matrices

What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not ...

**0**

votes

**1**answer

94 views

### Are they the $N\times N$ random matrices with real or complex entries asymptotically free?

The $N\times N$ random matrices with real or complex entries are generalizations of non-hermitian gaussian ensembles, also known as Girko ensemble: the entries are independent and identically ...

**4**

votes

**1**answer

243 views

### Confusion about Montgomery's Pair Correlation Conjecture

This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4
To set up the ...

**1**

vote

**1**answer

134 views

### What are the disadvantage and advantages of the moment method and the resolvent method in Random Matrix Theory?

I am learning random matrix theory . I am aware that the most popular successful techniques for obtaining the limiting spectral measure of large Hermitian random matrices are the moment method and ...

**27**

votes

**3**answers

6k views

### What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...

**2**

votes

**2**answers

101 views

### Spectrum of finite-band random matrices?

Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...

**4**

votes

**1**answer

90 views

### Haar unitaries with constraints

Given that one can sample unitaries from the Haar measure over $U(n)$ (as in F. Mezzadri, Notices of the AMS 54 (2007), 592-604), how can one sample from the uniform distribution over the following ...

**5**

votes

**5**answers

340 views

### Reviews of Probability in High Dimension not by Van Handel

I'm completely in love with Ramon van Handel's lecture notes Probability in High Dimension and I would like to find more learning resources. Lecture notes or reviews would be ideal as anything in this ...

**0**

votes

**1**answer

69 views

### How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem
Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$
Find the marginal distribution of each eigenvalue, using whatever you can.
Background
In my ...

**1**

vote

**1**answer

127 views

### Pair of vectors multiplied by a random matrix and its inverse transpose are distributed randomly up to their dot product

Given arbitrary nonzero vectors $\vec{x}_1, \vec{y}_1, \vec{x}_2, \vec{y}_2 \in \mathbb{Z}^{n}_p$ ($p$ prime) with $\langle x_1, y_1 \rangle = \langle x_2, y_2 \rangle$, I am trying to show that: $(...

**0**

votes

**0**answers

61 views

### Conditonal convergence implies convergence?

Note : All measures below are probability measures.
Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$.
Actually,...

**0**

votes

**0**answers

90 views

### A general question on random matrices

To give some context, suppose you have a sequence of random variables $X_n$ that you know weakly converges to a random variable $X$. Suppose you also have a sequence of random variables $Y_n$. Then, ...

**2**

votes

**0**answers

24 views

### Confused about the generality of statements regarding rank one deformations of Wigner matrices

I'm interested in the behavior of the largest eigenvalue of random $n \times n$ Hermitian matrices of the form $A^{n} = \frac{1}{2}(Q^{n}+(Q^{n})^{\dagger})$ where the real and imaginary components of ...

**3**

votes

**0**answers

112 views

### A guide to reading Alice Guionnet's Random Matrix book

I was reading Alice Guionnet's book "Large Random Matrices: Lectures on Macroscopic Analysis". I would need some help in understanding the author intends to do in Part III, "Matrix Models".
In the ...

**7**

votes

**2**answers

391 views

### Generating function of $SO(N)$ random matrix

I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute
$$
Z_N[J]=\int dM e^{{\rm Tr} (J^T M)},
$$
where $dM$ is the $SO(N)$ Haar measure, and $J$ is an ...

**6**

votes

**2**answers

292 views

### Probability of a large random integer Matrix to have zero determinant

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