# Questions tagged [random-matrices]

For questions concerning random matrices.

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### Probability of row sum zero random matrix being PSD

Consider $A$ to be a symmetric random matrix. When $i<j$, $A_{ij}=1+\sigma W_{ij}$ where $W_{ij}$ are standard gaussian variables, $A_{ij}=A_{ji}$ and $A_{ii}=\sum_jA_{ij}$. The $\sigma$ is a ...
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### Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns

Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i....
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### Free cumulants of Gaussian matrices with independent entries

Setup: Remark 5.1 in this work states that when $X\in\mathbb{R}^{n\times p}$ have i.i.d. $N(0,1/p)$ entries, the limit distribution of $XX^\top$ is the Marcenko-Pastur law, with limiting rectangular ...
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### Expected Value Largest Eigenvalue of a Random matrix

For a symmetric matrix whose entries are chosen uniformly at random $[-1,1]$ how do I find the expected value of the maximum eigenvalue of the matrix. As far as I understand finding this expected ...
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### What is the expected number of walks with length 𝑘 in Erdős–Rényi random graph?

Let $G(N, p)$ be a directed Erdős–Rényi random graph with edge probability $p$. Let $W_k$ denote the number of walks (potentially with repeated vertices, or repeated edges) of length $k$ beginning at ...
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### Eigenvalue density of a matrix whose kth moment is equal to 1/k

This is an exercise from the book "A First Course in Random Matrix Theory" by Potters and Bouchaud. We're interested in finding the eigenvalue density $\rho(\lambda)$ of a large random ...
1 vote
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### Why are square Bernoulli matrices invertible with high probability?

Given a matrix $A\in\mathbb{R}^{m\times n}$ with entries of $A$ being sampled i.i.d. from $\text{Bernoulli}(\alpha)$, where $\alpha\in(0,1)$ is a fixed constant. This paper (2nd sentence below ...
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### Sum of (non-free) symmetric and asymmetric random matrices

Consider random matrices $X,Y\in \mathbb{R}^{N\times P}$, i.e., $X_{ik}\sim{\cal N}(0,1/N)$ i.i.d. and likewise for $Y$. The eigenvalues of the symmetric covariance matrix $X^TX$ reside on the real ...
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### A question in operator norm of random matrix by epsilon net in Tao's RMT notes

I am reading Tao's Random matrix notes, specifically Proof of Corollary 2.3.5. Corollary 2.3.5. (Upper tail estimate for iid ensembles). Suppose that the coefficients $\sigma_{ij}$ of $M$ are ...
1 vote
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### SVD distribution of linearly transformed Gaussian ensemble

The joint pdf of the singular values of the $m \times n$ Gaussian ensemble $X = x_{i,j}$, where the $x_{i,j}$'s are independent Normal(0,1) samples and the eigenvalues of the associated Wishart ...
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### Asking if a diagonal Gaussian matrix satisfies a compressed sensing assumption

This is regarding the compressed sensing problem. In the problem $y=X\beta$ where $y\in\mathbb{R}^{n}$, $X\in\mathbb{R}^{n\times p}$, and $\beta\in\mathbb{R}^{p}$, it is known that a random matrix $X$ ...
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### Asking if a Bernoulli matrix and its variants satisfy the Restricted Isoperimetric Property (RIP) condition

This is regarding the compressed sensing problem that I recently learned about. In the problem $y=X\beta$ where $y\in\mathbb{R}^{n}$, $X\in\mathbb{R}^{n\times p}$, and $\beta\in\mathbb{R}^{p}$, it is ...