For questions concerning random matrices.

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2answers
37 views

Generating a random matrix with all eigenvalues equal to one

I want to generate a random matrix whose all eigenvalues are equal to one. How can I do it? I know one method to generate matrices with given eigenvalues is to generate a random orthogonal matrix $Q$ ...
2
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0answers
25 views

Symplectic Eigenvalues of Wishart Matrix

We work over the reals. Fix a dimension $n$ and a symplectic form $\Omega$. This is a $2n \times 2n$ matrix s.t. $\Omega^2 = -1$. A symplectic matrix is a matrix $S$ such that $ S^T \Omega S = ...
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1answer
25 views

Is there a way to generate random matrix that satisfying a give Linear Matrix Inequality?

The title is self-clear, I think. For example, we could generate a $n \times n$ random rank 1 positive semidefinite matrix by generate a random vector $x$ and $x'x$ will be the random matrix we want. ...
3
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0answers
26 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
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3answers
34 views

$A$ is a doubly stochastic matrix, how about $A^TA$

I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix. However, the paper says $A^TA$ is symmetric and stochastic. Since $A^TA$ is symmetric, if $A^TA$ is ...
4
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2answers
179 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with a normal distribution with the mean 0 and variance $m$) to its diagonal. A is a square ...
0
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0answers
33 views

random matrix perturbation in linear system

Let $\Phi$ be a $m\times n~ (m<n)$ matrix whose entries are i.i.d. normal Gaussian variables, i.e., $\Phi_{i,j}\sim \mathcal{N}(0, 1)$. Project a vector $\hat{x}$ to $\Phi$ we have $y=\Phi ...
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0answers
18 views

Semicircle Law with identically distributed but not independent matrix entries

I was wondering if there was some generalization of Wigner's semicircle law for Hermitian matrices with Gaussian distributed entries with bounded covariance (i.e., the distribution of the eigenvalues ...
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0answers
11 views

Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
0
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0answers
12 views

pseudo-Wishart distribution with shifted rows

I have a problem and I don't know where to start finding a solution. The problem is that I have a vector of i.i.d normal random variables such that, ...
2
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0answers
24 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
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2answers
49 views

Probability of a random matrix to be invertable.

Suppose that $x_{ij},i=1,2,\ldots,n;\,j=1,2,\ldots,m$ are independent and identically distributed continuous random variables. What is the probability that the group of vectors ...
2
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0answers
32 views

Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
0
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0answers
8 views

Stochastic independence of columns of projection matrix to the rest of the columns of a random matrix

First let me describe the setting of the problem. I have a random matrix $A\in \mathbb{R}^{m\times n},\ (m<n)$ with $a_{ij}\sim \mathcal{N}(0,I)$ i.i.d. Let there be a given set of $K (K<m)$ ...
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0answers
18 views

Linear affine random dynamical systems - positive Lyapunov index proof check?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb ...
1
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0answers
26 views

Matrix $A$ has random integer entries. Probability to get a matrix of integer elements after $A$ has been echelon row reduced?

Let $A$ be an $n \times m$ matrix with integer elements uniformly randomly chosen between $-k$ and $k$. What is the probability to get a matrix of integer elements after $A$ has been reduced in ...
2
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0answers
26 views

Prove that the eigenvalues of a random matrix of this form, are invariant regardless of the value of the exponent $s$.

Consider the sequence: $$\frac{\sum _{k=1}^1 \frac{1^s}{k^s}}{1^s},\frac{\sum _{k=1}^2 \frac{2^s}{k^s}}{2^s},\frac{\sum _{k=1}^3 \frac{3^s}{k^s}}{3^s},...,\frac{\sum _{k=1}^n \frac{n^s}{k^s}}{n^s} ...
1
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0answers
42 views

Product of Wishart and inverse Wishart distributions

Let $$ X \sim \mathcal{W}_{q} (n, \Sigma) \; \; n > q$$ and $$ Y \sim \mathcal{W}^{-1}_{q} (n, \Sigma^{-1}) \; \; n>q$$ Where $\mathcal{W}$ denotes the Wishart distribution and ...
0
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0answers
21 views

relationship of semicircular and circular elements, free fock space

I am trying to understand an argument in the paper Limit laws for Random matrices and free products by Dan Voiculescu, p 212. Let $\mathscr{T}\left(H_{n}\right) := ...
0
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0answers
29 views

on the spectral norm of the inverse of a random matrix

I have the following problem. A is a sum of independent random matrices and B is some fixed positive (semi-)definite matrix. I'm interested in a bound on the expectation of the operator norm $\lVert ...
0
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1answer
37 views

Proof that $\text{span} \{v_1,…,v_k\} \cap \text{ker}(T) = \{0\}$ if $\{v_1,…,v_k\}$ are vectors in general linear position.

The problem set up is as follows: Let $\omega^{(i)} \in \mathbb{R}^n$, for $i=1,2,...,k$, $k \le n$, be i.i.d. random vectors (whose distribution is irrelevant). Also, let $A \in \mathbb{R}^{m \times ...
3
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1answer
50 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
0
votes
1answer
143 views

Generating a random matrix with prescribed conditions

I need to uniformly generate a random matrix $X$ with positive integer entries satisfying a number of prescribed conditions: The matrix dimensions are prescribed, say $m\times n$ For each row, the ...
0
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1answer
35 views

The difference between a matrix valued random variable and an $n \times p$ matrix of data

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some ...
0
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0answers
20 views

Generating 2D random matrix with desired features

I try to generate a random demand matrix (from/to) with several features for my research. The point is that the total amount of demand is in decreasing order for the demand points (1 to N). For any ...
1
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1answer
39 views

Distribution of a matrix product $\mathbf{a}^{H}\mathbf{H}\mathbf{b}$

Could someone help prove the following: I have two independent random vectors $\mathbf{a} \in \mathbb{C}^{M \times 1}$ and $\mathbf{b}\in \mathbb{C}^{N \times 1}$. Both $\mathbf{a}$ and $\mathbf{b}$ ...
1
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0answers
32 views

distribution of quadratic form of jointly normal random variables?

I need to derive the distribution of the random variable $\frac{W'(I-1(1'1)^{-1}1')ZZ'(I-1(1'1)^{-1}1')W} {Z'(I-1(1'1)^{-1}1')Z}$ , where $(Z, W)'$ ~ $N(0, I), \,Z=(Z_{1}, ..., Z_{n}), \,W=(W_{1}, ...
2
votes
1answer
48 views

Marcenko Pastur and smallest eigenvalue of a random matrix

I have a matrix with (supposedly) i.i.d. standard normal entries of dimension 2500*50000. When I find the singular values of this matrix, I am getting the smallest singular values to be very close to ...
0
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0answers
12 views

signal variance

I read a section about $H_2$ control. And one section is as following: It seems that the context assumes 1. $w(t)$ is a vector. i.e. $w(t) = [w_1(t) ...w_n(t)]^T$ 2. Each entry is zero mean. ...
2
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1answer
58 views

Do the eigenvectors of a random orthogonal matrix have Haar measure?

For orthogonal $Q$ with Haar measure, does the group of unitary matrices $U$ which diagonalize $Q=U\Lambda U^H$ have Haar measure? I'd be happy to know any answer, even if it's only true for certain ...
0
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0answers
21 views

Gaussian Measure for Random Matrix

I am doing physics and do not have enough mathematical background. so the question may be trivial, I apologize for that. any help would be highly appreciated. my question is: How does Random Matrix ...
0
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0answers
38 views

Uniformly randon invertible matrix over $\mathbb{Z}/q$

Is it possible to generate uniformly random invertible matrices over $q$-element finite ring $\mathbb{Z}/q$, where $q=p^n$, $p$ prime and $n \ge 2$, without try-and-repeat method? In other words, ...
0
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0answers
29 views

Are Kronecker products of Gaussian vectors almost surely linearly independent?

This is a question about the rank of a random matrix. Pick $n, p$ two integers (typically, $n > p$). Generate $n$ random vectors $y_1, \ldots, y_n$ in $\mathbb{R}^p$ following a Gaussian ...
1
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0answers
93 views

Pseudoinverse - Interpretation

An $m$-dimensional (column) vector $y$ is defined as follows: $Ay=x+v$, where $A$ is an $m*n$ matrix with $m<n$ (and full row rank), $x$ is an $m$-dimensional column vector of constants and $v$ ...
0
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1answer
48 views

expectation of matrices with random components

Let's say I have a matrix where some of the components are random variables. From Wikipedia, the expectation of the matrix is simply the expectation of the individual components. But, beyond that, ...
1
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0answers
20 views

Does concentration of measure inequality for a random matrix is enough to determine bounds on extremal singular value tail probabilities?

First of all I do not know if this even qualifies to be a proper question, but I have a rather trivial doubt which I somehow could not resolve for the past few hours. So here it is. Let $A\in ...
0
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0answers
10 views

fix a line for operation on matrices

operations on matricies determinant Why does a line at least must be fixed to to operations on matrices for their determinant calculation? I understand that if it was allowed not to do such a ...
0
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0answers
18 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
0
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0answers
21 views

Big O p Question about Eigenvalue of Random Matrix

Suppose $S_1, S_2, \dots$ are a sequence of random symmetric matrices in $\mathbb{R}^{d\times d}$. Suppose we know that $|\lambda_\max(S_n)| = O_p(b_n)$ and also that $|\lambda_\min(S_n)| = O_p(b_n)$ ...
0
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0answers
20 views

Distribution of $Mx$ where $M$ is a Haar matrix and $x$ is a unit vector

I have a $n\times N$ random matrix $R$ with entries i.i.d. standard Gaussian. $n<N$. Then the matrix $M=(RR^T)^{-1/2}R$ is Haar distributed, i.e. has orthonormal rows and uniformly random ...
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0answers
27 views

Qualitative properties of eigenvalues that can be inferred from matrix structure?

I am doing a linear stability analysis of a 6-dimensional system, what I want to know is if the system is stable at numerically solved steady states by looking at the eigenvalues of the jacobian ...
4
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0answers
46 views

How to prove the following about eigenvalues

Let $\mathbf{M} = [m_{ij}]$ be a symmetric matrix of size $m\times m$ of real elements. Let $\mathbf{A} = [a_{ij}^R + ia_{ij}^I]$ be a random Hermitian matrix whose elements have variance, $\sigma^2$, ...
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0answers
46 views

distribution of eigenvectors of a random matrix

Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a random hermitian matrix. Assume that the eigenvalues of this matrix have continuous probability distribution. 1.Can we say that the eigenvectors ...
1
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0answers
21 views

Eigenmatrix of a Wishart Matrix

A lot of papers mention statements of the form "The Eigenmatrix of a Wishart matrix are uniformly distributed on a sphere of $P$ dimensions" but I am having a hard time finding the conditions under ...
3
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1answer
25 views

Expected value of $X(X'X)^{-1}X'$ if X has normally distributes entries

Suppose that each entry in $n$ by $p$ matrix $X$ has standard normal distribution $\mathcal{N}(0,1)$. I am interested in finding the proof that $\mathbb{E}(X(X'X)^{-1}X') = \frac pn \cdot I_n$, ...
2
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0answers
32 views

Is it possible to show that $E\left\|\sum_{i=1}^nX_i\right\|^2_s\leq Mdn$?

Let $X_i\in\mathbb{R}^{d\times d}$ be i.i.d. random matrices with zero-mean elements and variance bounded by some $C$. Then $$E\left\|\sum_{i=1}^nX_i\right\|^2_s\leq ...
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0answers
12 views

(bounds on) mean of correlation matrix

Let $C\in\mathbb{R}^{p\times p}$ be a correlation matrix, that is a positive (and thus symmetric) definite matrix with main diagonal elements equal to $1$ and off-diagonal entries in the interval ...
0
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0answers
27 views

Third moment of Wishart matrix

I am looking for a closed form formula for the third non-central moment of a Wishart matrix: For $(s_{ij})\sim \operatorname{Wishart}_p(1,I_p)$, what is $\mathbb{E}[s_{ij}s_{stj}s_{kl}]$? I looked ...
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0answers
33 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
0
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0answers
20 views

Regarding distribution of product of matrix and its transpose

Let $\mathbf{H}$ be a matrix of size $m\times n(n>m-1)$ where each column of the matrix indicates an $m-$ dimensional measurement. and there are $n$ measurements in all. Let us assume that ...