Questions tagged [random-matrices]
For questions concerning random matrices.
899 questions
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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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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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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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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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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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Change of Variable in Probabilty Distribution
In the context of random matrix theory, consider the following scenario:
Let $M$ be a $2 \times 2$ real symmetric matrix, where the matrix elements are Gaussian distributed with the probability ...
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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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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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Conjugation action of Haar distributed matrices
Let's take the group $\operatorname{SU}(N)$.
It acts on $\mathfrak{su}(N)^* \cong \mathfrak{su}(N)$ by the conjugation action
$X \mapsto A^\dagger X A$, where $X \in \mathfrak{su}(N)$.
Imagine now ...
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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 ...
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Matrix-Free Stochastic Approximation
Let $C: \mathbb{C}^m \to \mathbb{C}^m$ be a sparse Hermitian operator that I only know via matrix-vector products. Using Chebyshev filtering, I obtain the k highest eigenvalues $\lambda$ and ...
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Marginalization of Wishart distribution
This is question I posted years ago on mathoverflow and got no answers: https://mathoverflow.net/questions/398216/marginalization-of-wishart-distribution
Consider the following Wishart distribution
$$
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Laplacian in Random Matrix Theory
I have a limited knowledge of the Laplacian operator and matrix calculus, but I would like to know how to evaluate $$\Delta\operatorname{Tr}(A^2)$$ where $A$ is an $n\times n$ matrix. I’ve been ...
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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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