For questions concerning random matrices.

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1answer
74 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 ...
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1answer
25 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 ...
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16 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 ...
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1answer
32 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 $\mathbf{b}$ ...
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24 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
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1answer
32 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 ...
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11 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
46 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 ...
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18 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 ...
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33 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, ...
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22 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 ...
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81 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 $v$ ...
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1answer
31 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, ...
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0answers
17 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 ...
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9 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 ...
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16 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 ...
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16 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 $|\lambda_\min(S_n)| = O_p(b_n)$ ...
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15 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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26 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 ...
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45 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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34 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 ...
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0answers
19 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 ...
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21 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 $\mathbb{E}(X(X'X)^{-1}X') = \frac pn \cdot I_n$, ...
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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
10 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 ...
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15 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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27 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 ...
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0answers
16 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 ...
2
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1answer
46 views

Joint distributions

Let $X, Y$ be continuos random variables of densities $f_X, f_Y$. Let $Z = \begin{pmatrix} X \\ Y \end{pmatrix}$. When is $Z$ continuos? And in this case, how to express its density with respect to ...
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1answer
27 views

Spectral norm of submatrices of a matrix with bounded spectral norm and maximum-entry

Let $A(t)=A$ be a symmetric, positive-definite matrix in $\mathbb{R}^{p\times p}.$ Suppose that the maximum-magnitude entry of $A_t$ has magnitude bounded above by $f(t)$. Suppose also that the ...
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0answers
56 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
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17 views

Eigenvalues/eigenvectors distribution of random tridiagonal matrices

Let $A\in\mathbb{R}^{n\times n}$ be a tridiagonal matrix with zero mean i.i.d. entries $a_{ij}$. (Notice that $A$ is not required to be symmetric). There exist general results on the limit ...
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0answers
26 views

Find rule generating seven 6-tuples (k=0,…,6) composed of two sets of consecutive relatively prime fractions

I have the seven 6-tuples listed below, functions of k (0,…,6), composed of four consecutive relatively prime fractions involving fifths, and two involving sixths. I would like to find a rule that ...
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0answers
51 views

Lower Bound on Expectation of Operator Norm

I've been working through Terence Tao's text on random matrices, and there's a step in a proof that I am having trouble with. We want to show Proposition 2.3.19. The assumptions are $M$ symmetric ...
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15 views

Complex Covariance matrix

I computed the covariance matrix of a 2 x 2 MIMO channel \mathbf{H}, which is modeled as a complex Gaussian channel with zero mean and \mathbf{\Sigma} covariance matrix. The covariance matrix I got is ...
4
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0answers
38 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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0answers
50 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
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13 views

How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
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1answer
29 views

When does the Singular Value Decomposition fail?

Does the singular value decomposition ever not work? The statement of the associated theorem, here from wikipedia: http://en.wikipedia.org/wiki/Singular_value_decomposition#Statement_of_the_theorem is ...
3
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1answer
38 views

How does additive noise change the SVD

For matrix $M$ with SVD $M=U\Sigma V^*$ and random matrix $A$, what is the SVD of $M+A$? That is, how will $A$ change the singular values and vectors of $M$? Let's even say that the entries of $A$ ...
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27 views

Distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
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14 views

Distribution of AYB in terms of distribution of Y

Let $A$ and $B$ be two random orthogonal matrices and let $Y$ be a random diagonal matrix. The distribution of $Y$ is known to be $p_y$. How can we express the probability distribution of $X = AYB$ in ...
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46 views

Concerning the problem of finding the number of invertible nxn random {1,0} matrcies

In a few more words, if we look at the space of all nxn matrices (over a field of characteristic 0) with only 1 or 0 as an element in them ("binary matrices"), how many of them are invertible for each ...
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2answers
55 views

iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
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0answers
22 views

Functional differentiation in Mehta's “Random Matrices”

I'm trying to understand a bit in this book about functional differentiation, which I don't know much about. According to Wikipedia, $\delta F=\int d^n\boldsymbol{r}\frac{\delta F}{\delta ...
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0answers
42 views

An analytic characterization of eigenvalues of a Hermitan matrix.

If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. (where we have $\lambda_i > \lambda_{i+1}$) for a vector $v$ let its component along the corresponding ...
3
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0answers
34 views

How to generate random symmetric unitary matrices “close” to a given matrix?

I want to perform Monte Carlo simulation for the analysis of a circuit problem, where the generation of random symmetric unitary matrices "close" to \begin{equation}T=\left[ {\begin{array}{*{20}{c}} ...
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33 views

Product of random binary vector with random binary matrix in GF(2)

Suppose we have a binary vector $f$ with dimensions $1×l$ such that each entry in the vector is generated independently with propability $q$ of being $1$. And we have a binary matrix $G$ with ...
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1answer
104 views

Magic Squares with Random Numbers

I'm trying to solve a problem related to Magic Squares. The thing is: Given a list of n numbers, I need to answer if it is possible to create a magic square with them. These numbers are random (no ...
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1answer
22 views

Sampling a matrix of an AR model

Let us consider a dynamic system $x_t = A x_{t-1}+v_t$ where $v_t$ is multivariate normal noise with zero mean, i.e. $v_t\sim\mathcal{N}(0,\Sigma)$ and $A$ is a matrix. As far as I know, for some $A$, ...