Questions tagged [random-matrices]
For questions concerning random matrices.
898 questions
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Eigenvector of GSE matrix with right properties
I encountered a weird problem when trying to study a few properties of eigenvectors of matrices sampled from the Gaussian symplectic ensemble (GSE). I have encountered this while trying to understand ...
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Commutant of random linear combination of matrices
I'm not too familiar with random matrix theory so I cannot find a suitable reference for this question.
Consider a set of matrices $\{A_i\}_{i=1}^k\subseteq M_{d\times d}$ over the complex field and ...
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How to Construct an Integer Matrix Whose Submatrices Often Generate $\mathbb{Z}^k$
I encountered the following problem when I tried to design a cryptographic protocol.
Suppose we want to construct a matrix $\mathbf{G} \in \mathbb{Z}^{n \times k}$ with $k \leq n$ such that:
$\mathrm{...
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Hoeffding bound for random matrices proof question
The following is from High-Dimensional Statistics: A Non-Asymptotic Viewpoint by Wainwright.
Throughout, all matrices will be symmetric in $\mathbb{R}^{d \times d}$. For a matrix, let $\lVert A \...
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Randomness of Columns of a Matrix
Suppose I have a set of fixed vectors $\{f_n\}_{n=1}^N, \{g_n\}_{n=1}^N \subseteq \mathbb{R}^d$. If I want to build a matrix $M$ of size $d \times N$ with the following rule, let $M_i$ denote the $i$...
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Expected value of squared determinants with Gaussian noise
Consider $Y = A + \varepsilon W$, where $A$ is a deterministic $n \times n$ matrix and $W$ has $N(0,1)$ i.i.d. entries.
This setup arises naturally in random matrix theory when studying denoising.
I'm ...
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Laplacian in Random Matrix Theory
I have a limited knowledge of the Laplacian operator and matrix calculus, but I would like to know how to evaluate $$\Delta\operatorname{Tr}(A^2)$$ where $A$ is an $n\times n$ matrix. I’ve been ...
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What is the probability that the determinant lies in the interval $[-9, 9]$?
Given the $3 \times 3$ matrix
$$A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{pmatrix}$$
where every $a_{ij} \in {...
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Estimating the error when averaging a function of a matrix over a collection of random matrices
In short, I want to understand how to estimate the error in calculating the average of a function on a random matrix. I expected to be able to use the standard error of the sample mean, but that hasn'...
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Expected value of the inverse of $C = \frac{1}{N}AA^{T}$
Let $A\in\mathbb R^{P\times N}$ be a matrix whose entries $a_{\mu k}$ are i.i.d. random numbers sampled from $\{-1,1\}$ with same probability $1/2$.
Let $C = \frac{1}{N}AA^{T}$. I want to calculate $\...
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Operator norm of a sum of random rank-1 matrices with uniform spherical factors
I am studying the operator norm (spectral norm) of the following random matrix.
Let $ u_i \in \mathbb{R}^p , v_i \in \mathbb{R}^q $ be independent random vectors, where each $ u_i \sim \mathrm{Unif}(\...
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Non-equivalence between intra-row/intra-column shuffles and elementwise shuffles on a square matrix
Background
Somebody asked a question on the Chinese Q&A website ZhiHu(知乎), which roughly translates to the following: given an $n\times n$ matrix $M$ consisting of $n^2$ distinct real numbers, ...
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Why are these squared singular values distributed as $\exp(-\pi X^2)$ with Gaussian $X$?
Let $x_1, x_2,\ldots,x_d$ be $d$ independent random vectors in $\mathbb{R}^d$ with entries sampled IID from the standard normal random variable $\mathcal{N}(0,1)$. Define the $d\times d$ random ...
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A Problem in finding expectation and variance of an $n \times n$ random matrix with Bernoulli $\left( \frac{1}{2} \right)$ entries.
Consider an $\large n \times n$ order matrix $\large M$. The $\large i,j$-th entries of the matrix $\large M$, let's say, $\large X_{i,j}$ is an i.i.d random variable ($\large \forall i,j$) following ...
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Bounding expected spectral norm of shifted Grammian of a Rademacher random matrix
Let $S \in {\Bbb R}^{k \times n}$ be a random matrix with independent entries
$$ {\Bbb P} \left[ S_{ij} = \pm \frac{1}{\sqrt{k}} \right] = \frac12 $$
I am interested in finding the tightest possible ...
