For questions concerning random matrics.

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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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26 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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13 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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29 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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7 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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11 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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24 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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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 ...
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1answer
38 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
18 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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43 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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12 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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1answer
25 views

Good book on random quadratic forms

I am studying some algorithms which are very much based on quadratic forms involving complex Gaussian Random vectors, something like this $ \vec{x}^* M \vec{x} $ where $x \in \mathbb{C}^{N \times 1}$ ...
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24 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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37 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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12 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 ...
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32 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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42 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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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
24 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 ...
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1answer
33 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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26 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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39 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
40 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
25 views

Compute expectation of eigenvalues of a random matrix

Is there any quicker way to compute the expectation of the eigenvalues of a random matrix than Monte Carlo method?
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21 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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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 ...
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30 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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29 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
78 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
21 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 $A$, ...
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21 views

Lower bound on the sum of rank of matrices

Consider $G_1,G_2$ are matrices of size $n\times n$. $G_1,G_2$ are independent. One can see it as every entry of them is drawn uniformly and i.i.d. from real line, so with probability one $G_1$ and ...
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53 views

Efficiently deleting 2s from a random NxM matrix

Edit: There were 2 important logic errors in the code below. They have been fixed! update: I still don't have an answer to this question, but I recently made a massive improvement to my current ...
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1answer
55 views

Why for random matrix one of eigenvalues is so big?

Here i have the image of eigenvalues for matrixes obtained with $rand(n,n)/sqrt(n)$, why one of the eigenvalues for every matrix is so big on real axis comparatively to the others? EDIT: i have ...
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16 views

How do you count a grouping without a sequential count?

So, rather than explaining how this problem pertains to the actual situation I think its easier and a great deal less work to give you a situation that you can visualize. Imagine a person who has ...
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2answers
153 views

Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky ...
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1answer
49 views

An algorithm of solving a non-homogeneous linear equation by random matrices

I'm looking for the proof of the following numerical algorithm. Suppose I want to solve a non-homogeneous linear equation \begin{equation} A x = b \end{equation} The matrix $A$ is non-invertible and ...
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33 views

Bound on Signal Amplitude for subspace methods (MUSIC, ESPRIT)

MUSIC and ESPRIT are methods that use subspace decomposition to identify signal Parameters. Subspace decomposition is achieved either by SVD or Eigen Value Decomposition. Subspace decomposition ...
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1answer
23 views

Effect of the nature of noise on the spectrum of a random matrix

Consider the following two equations $X = M + \eta_1$ $Y = M + \eta_2$ where, $X\in\mathrm{R}^{n\times n}$, ia a real random matrix with mean $M\in\mathrm{R}^{n\times n}$. $\eta_1$ is Gaussian ...
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Probablistic bound for $\|RR^TM\|$ for uniformly random orthonormal matrix $R$

I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ...
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22 views

What does one-cut random matrix mean?

I am quite new to random matrix theory and recently I encountered the so-called "one-cut random matrix model" and even "two-cut" in physics. So what exactly does it mean?
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Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
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1answer
72 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
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231 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
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0answers
107 views

Derivative of Cholesky decomposition

I would like to compute the derivative of the Cholesky decomposition, for example I have a matrix 2 x 2 R = 1 rho rho 1 where rho is a parameter, now I compute the Cholesky decomposition of ...
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2answers
56 views

Constructing 5 by 5 Unitary matrices

I am trying to construct an arbitrary 5 x 5 Unitary matrix. Any example will be appreciated.