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Questions tagged [random-matrices]

For questions concerning random matrices.

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Expectation of Non-Isotropic Projection Matrix

Let $\Pi$ be a projection matrix (i.e. $\Pi^2=\Pi$) derived from a random sample ($X$) which is from a $N(0,\Sigma)$ distribution. That is, $\Pi = X^T(XX^T)^{-1}X$. I am looking at figuring out $E[{\...
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Technique for finding expected value Weighted Ridge Regression Coefficients

Context: We would like to approximate a linear function $f(\mathbf{x})$ at the point $\boldsymbol{\xi} \in \mathbb{R}^D$ using samples of size $N$ around $\boldsymbol{\xi}$. Assume that the input data ...
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Solution to exercise 1.1.12 of Terence Tao's Random Matrix Theory notes

Tao's notes: https://terrytao.wordpress.com/wp-content/uploads/2011/02/matrix-book.pdf I understood Example 1.1.3. by explicitly writing down every element in hyperplane V. However, in Exercise 1.1.12 ...
Nao Tomita's user avatar
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Matrix function of a measurable map $f$ is again measurable?

Consider a measurable map $f: \mathbb{R} \to \mathbb{R} $. Define the matrix function as the induced map \begin{equation} f: H^n \to H^n \\ A = \sum_i \lambda_i e_i e_i^T \mapsto \sum_i f(\lambda_i) ...
2000mg Haigo 's user avatar
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Does $\text{Var}((AWy)^TAWy) \geq \text{Var}((AW_1y)^TAW_1y)$?

Suppose $W$ is an $n\times n$ random matrix with each entry i.i.d. $\sim N(0,1)$, let $A = (WDW^T +\lambda I_n)^{-1}$ where $D$ is a diagonal matrix with every entry $>0$, $I_n$ the idendity matrix ...
FreshSSS's user avatar
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Free probability version of Poincaré Separation Theorem

Suppose $A$ is a $d \times d$ real positive semi-definite matrix, and $U$ is a $d \times n$ semi-orthogonal matrix such that $U^\top U = I_n$. Define $B = U^\top A U$. The Poincaré Separation Theorem ...
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Exercise 3.3.3 (c) of High dimensional probability by Roman Vershynin

I have question Exercise 3.3.3 (c) from Chapter 3. Exercise 3.3.3 (Rotation invariance). Suppose $ G $ is an $ m \times n $ Gaussian random matrix, i.e., the entries of $ G $ are independent $ N(0,1) $...
chloe's user avatar
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63 views

Calculating trace of inverse of a random matrix

I am trying to calculate $\frac{1}{r}$Tr$((\mathbf{I}_r-B)^{-1})$ where $\mathbf{I}_r$ is the identity matrix and B is a random $r$ by $r$ matrix given by $B = O^{T} DO$, where $O_{n\times r}$ is ...
Aaradhya Pandey's user avatar
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What does the spectral norm of a Wigner matrix converge to when the variances are not renormalised?

It seems that it is well known that for a $NxN$ Wigner matrix - that is a matrix that is symmetric (or Hermitian, but I am only interested in the case where all the entries are real) and has i.i.d. ...
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For a given a non-hermitian random matrix, how large can you make its leading eigenvalue using only swaps across the diagonal?

Suppose $X$ is an $N\times N$ Ginibre random matrix. Each element of the matrix $x_{ij}$ is drawn independently from a Gaussian distribution with zero mean and variance $\mathrm{E}[x_{ij}^2]=\frac{1}{...
Lyle's user avatar
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How does the spectral radius of the Hessian of a box-constrained polynomial with random coefficients change with degree and dimension?

It seems that spectral radius of a symmetric random matrix with i.i.d. entries 0 mean and some assumptions on variance converges to $\sqrt(n)$ where $n$ is the number of dimensions: e.g. https://arxiv....
ufghd34's user avatar
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How to compute mean and covariance for repeated transformation of random matrices?

I have a random matrix $M$, which in practice is vectorized/flattened and represented as a multivariate Gaussian, so I know the mean and covariance of $M$. Consider the following system: $$ X_{t+1} = ...
Ralff's user avatar
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Smallest eigenvalue of matrix with random elements (non-central Wishart)

Suppose that $X \in \mathbb R^{d \times n}$ is a random matrix with independent entries, each of which follows the standard normal law $\mathcal N(0, 1)$, and that $M \in \mathbb R^{d \times n}$ is a ...
Roberto Rastapopoulos's user avatar
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Eigenvalues of Hermitian quaternionic matrices

This is a question related to the Gaussian symplectic ensemble in random matrix theory. I know that the eigenvalues of the GOE and the GUE are all real, and it seems to be that this should be the case ...
Luis's user avatar
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Density of product of random matrix with vector

Assume that we have a random matrix $M \in \mathbb{R}^{d \times d}$, and we multiply it by a vector $v$ that is constant (for convenience, you can assume that it's a unit vector). Is there a general ...
susami's user avatar
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Rank analysis of a random binary matrix within a random binary matrix

Let's say we have a matrix $A \in \mathbb{R}^{m, n}$ which is a random binary matrix where each entry draws from the Bernoulli distribution with $P(n=1) = p_a$ and $P(n=0) = 1 - p_a$. Similarly, we ...
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Jacobian Determinant of Orthogonal Conjugation of Matrices

I came across this problem while studying GOE Wigner matrices in random matrix theory. Note that a GEO matrix is a symmetric matrix whose off-diagonal entries are i.i.d with $\mathcal{N}(0,1)$ and ...
Jamal's user avatar
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2 votes
2 answers
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Rank of binary matrices: {0, 1} vs {-1, 1}

Let's say you have two matrices, $M_{\{0, 1\}}$, a binary matrix with entries in $\{0, 1\}$, and $M_{\{-1, 1\}}$, a binary matrix with entries in $\{-1, 1\}$. Both matrices are identical, except where ...
user3667125's user avatar
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1 answer
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Unitary invariance of the Gaussian Unitary Ensemble

I'm stuck with a property of a GUE matrix. In the textbooks it is claimed to be "obvious" but I can't figure out why. Suppose we have a GUE matrix $G$ and a unitary matrix $U$. Then $UGU^{-1}...
Bla Blub's user avatar
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Expected rank of a random binary matrix with Bernoulli probability p?

Let $M \in \mathbb{R}^{m,n}$. The entries are in {$0, 1$} (with the value $1$ having probability $p$, and the value $0$ having probability $(1-p)$). What is the expected rank of $M$? Follow-up: how ...
user3667125's user avatar
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How to calculate the variance of this random variable?

$\theta$ is a random variable whose distribution is a normal distribution with mean $m_0$ and variance $\sigma_0^2$. Let $u$ be a $k\times1$ random vector following a normal distribution where the ...
Ypbor's user avatar
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Distribution of $\varepsilon^T P_\Theta \varepsilon$ where $\varepsilon, \Theta$ are jointly Gaussian and $P_A = A (A^T A)^{-1} A$ orth. projection

Let $n \geq k$. Let $\varepsilon \in \mathbb{R}^{n}$ and $\Theta \in \mathbb{R}^{n \times k}$ with $(\varepsilon, \Theta) \sim \mathcal{MN}((0, \Theta_0), \mathrm{Id}, \Omega)$, where $\mathcal{MN}$ ...
M. Londschien's user avatar
5 votes
3 answers
180 views

Computing Lyapunov Exponents of an Example of Avila and Bochi

In Artur Avila and Jairo Bochi's lecture notes (see here: http://mat.puc-rio.br/~jairo/docs/trieste.pdf) in section 3.1 they deal with Lyapunov exponents of products of random i.i.d. matrices. Let $\{...
Raul Fernandes Horta's user avatar
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33 views

Lower bound of $\frac{\|(\mathbf X \otimes \mathbf X^\top)\theta\|_2^2}{np}$

According to Theorem 7.16 of High-Dimensional Statistics: A Non-Asymptotic Viewpoint (M. Wainwright, 2019), we know that for $\mathbf X\in\mathbb R^{n\times p}, X_{ij}\overset{iid}{\sim}N(0,1),$ there ...
Jasper Cha's user avatar
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Measure transport by a random matrix

I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are ...
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Bounding absolute value of centered row sum of adjacency of Erdos-Renyi graph

I am trying to understand the formula (5.14) in this paper, explicitly, the goal is to upper bound $$\sum_{i\neq j}|a_{ij}-p|$$ where \begin{equation} a_{ij}=\begin{cases} 1,\text{ with ...
chloe's user avatar
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How to bound a sum of absolute value of centered Randemacher variables

Consider \begin{equation} A_{i}=\begin{cases} 1,\text{ with probability } 1-p\\ -1,\text{ with probability } p\\ \end{cases}\,. \end{equation} and $p<1/2$. What would be the ...
chloe's user avatar
  • 1,052
2 votes
1 answer
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Uniform bound involving trace of random matrix

For each ${n}$, let ${A_n = (a_{ij,n})_{1 \leq i,j \leq n}}$ be a random ${n \times n}$ matrix (i.e. a random variable taking values in the space ${{\bf R}^{n \times n}}$ or ${{\bf C}^{n \times n}}$ ...
shark's user avatar
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2 votes
1 answer
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Integration by substitution in Selberg's Integral

I am reading the article "Hilbert--Schmidt volume of the set of mixed quantum states" (https://arxiv.org/abs/quant-ph/0302197). I do not understand the step in which we start from (4.2) and ...
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A question on positivity of eigenvalues for a matrix with some random chosen entries

Let $0<c<1$. Is it possible to construct a symmetric $n\times n$ matrix $A=(a_{ij})$ with $a_{ii}=1$ for all $1\leq i\leq n$ and $a_{ij}\in ${$1, c$} for all $i\neq j$ for some $n\geq 1$ such ...
ougao's user avatar
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+50

bounded density for the determinant of a GOE

Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
kaleidoscop's user avatar
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Expectation of Random Projection Matrix

Suppose $x_i \sim N(0, I_p)$ for $i\in[n]$ where $n < p$. Let $X = (x1,...,x_n)^T$ What is $E[X^T(XX^T)^{-1}X]$? Clearly $(XX^T)^{-1}$ is Wishart. I have only seen Normal Inverse Wishart and not ...
Jake Freeman's user avatar
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Computing Haar measure of matrices sampled from SO(n)

I am looking to sample uniform matrices from SO(n). I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
magnesium's user avatar
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2 votes
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Introductory books in Random Matrix Theory for Riemann zeta function

In many papers when studying Riemann zeta function (like Alternate Hypothesis, or pair correlation conjecture) I faced with terms like "gaussian unitary ensemble" or "random matrix ...
Ali's user avatar
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Random measure and empirical spectral distribution

I have a question about the application of the definition of a random measure on the empirical spectral distribution. Let $(X , \mathcal{B})$ be some measurable space and let $(\Omega, \mathcal{F}, \...
MathAccount12's user avatar
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Expectation of $XX.T$ vs Expectation of X * Expectation of X.T

I need to understand what follows: Given $X \in \mathbb{R}^{T \times n_0}$ $W \in \mathbb{R}^{n_0 \times n} : W_{ij}\sim \mathbb{N(0, \frac{1}{n_0})} \space i.i.d.$ $\sigma : \mathbb{R} \rightarrow \...
fabianod's user avatar
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4 votes
0 answers
107 views

Mean log determinant of a random Gaussian matrix

I wish to calculate the mean log determinant of a random Gaussian matrix, defined by the following definite integral (for $a, b, \sigma\in\mathbb{R}$): $$ I(a, b, \sigma) = \int \frac{d^{n \times n}X}{...
Uri Cohen's user avatar
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0 answers
24 views

Johnson Lindenstrauss lemma for positive entries matrix

I read about the Johnson Lindenstrauss lemma, which (roughly) states that if $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{k \times n}$ is a "random matrix", then with high probability $\|x\|...
Johana T's user avatar
1 vote
1 answer
153 views

Convergence/divergence of the operator norm for random sign matrices

For each ${n}$, let ${A_n = (a_{ij,n})_{1 \leq i,j \leq n}}$ be a random ${n \times n}$ matrix (i.e. a random variable taking values in the space ${{\bf R}^{n \times n}}$ or ${{\bf C}^{n \times n}}$ ...
shark's user avatar
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1 vote
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68 views

Joint moment $\tau(XYXYXY)$ in terms of moments of $X$,$Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
1 vote
0 answers
60 views

Bounding $v_1^\top D v_1$ where $v_1$ is the top eigenvector of a symmetric random matrix (Wigner matrix)

Suppose $A\in \mathbb{R}^{n\times n}$ is a symmetric random matrix, with iid centered Bernoulli non-diagnonal entries $A_{ij} \sim \mathrm{Bernoulli}(p) - p$, for some $p \in (0,1)$, and zeros on its ...
JasonShaw's user avatar
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25 views

Universal concentration of random matrix

I am studying a family of random matrices and observe a universal concentration behavior which I would like to understand. Let $U$ be an $N \times N$ random unitary drawn from the Haar measure, and $Z ...
nervxxx's user avatar
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0 answers
80 views

Joint probability distribution of matrix elements of Gaussian orthogonal ensemble (GOE)

The Gaussian orthogonal ensemble of $N\times N$ symmetric matrices is often defined as a matrix whose diagonal elements are drawn from the Gaussian distribution $N(0,1)$ and off-diagonal elements from ...
Landuros's user avatar
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12 votes
2 answers
336 views

Why is $AA^TAA^TAA^TAA^T$ larger than $AAAAA^TA^TA^TA^T$ for Gaussian $A$?

Suppose $A$ is a $d\times d$ matrix with IID standard normal entries. Plots below compare value of $f(AA^TAA^TAA^TAA^T)$ and $f(AAAAA^TA^TA^TA^T)$ using 3 standard Schatten norms for $f$ and the ...
Yaroslav Bulatov's user avatar
2 votes
2 answers
231 views

Value of $E_{A,y}[\cos(A^T y,A^{-1} y)]$ for Gaussian $A,x,y=Ax$

For $d\times d$ matrix $A$, $x\in \mathbb{R}^d$ with IID standard normal entries and $y=Ax$, I'm interested in the the following quantity where $\cos(u,v)$ refers to cosine similarity: $$E_{A,x}[\cos(...
Yaroslav Bulatov's user avatar
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0 answers
16 views

Moment method / genus expansion for random matrices with i.i.d. entries

Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$ ...
Panda Jonas's user avatar
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21 views

Problem with Wigner matrix

My problem: In the following, all the variables considered $X_{i j}, X_{i j}^{(k)}$ are: i.i.d. centered with variance 1 and fourth finite moment for $i \leq j$. We set $X_{i j}=\overline{X_{j i}}, ...
Pipnap's user avatar
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Orthogonal transformation of Heteroskedastic matrices

Consider two $N \times N$ dimensional real matrices $A$ and $B$. $A$ is a diagonal matrix with all non-zero elements taken from a real Gaussian distribution with mean $\mu = 0$ and variance $\sigma = \...
BB_'s user avatar
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0 answers
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Random bounded triangular $T_n$ s.t. $Var[T_nS_nT_n^\top]\to 0$ for nonrandom psd $S_n$. Does $(T_n-\bar T_n)S_n\to^P0$ for some nonradom $\bar T_n$?

Let $T_n$ be a sequence of square random matrices with $T_n$ lower triangular with $diag(T_n)=(1,1...,1)$ and $S_n$ a sequence of deterministic symmetric psd matrices. All matrices are in $R^{d\times ...
jlewk's user avatar
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1 vote
1 answer
86 views

degeneracy of even-dimensional "pullback" of symplectic form

Suppose $\omega(u,v) = \left\langle Ju, v\right\rangle$ is the canonical nondegenerate symplectic form on $\mathbb{R}^{2N}$. Given a random left-orthogonal matrix $P\in\mathbb{R}^{2N\times 2n}$ (...
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