For questions concerning random matrics.

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9 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
15 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
27 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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0answers
25 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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0answers
12 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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0answers
33 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
24 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
16 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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0answers
20 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
37 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 ...
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0answers
26 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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0answers
17 views

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

Suppose we have a binary vector $f$ with dimensions $1√ól$ 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
61 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
18 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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0answers
16 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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0answers
51 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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0answers
54 views

Homework help, cramers rule

A company earns before-tax profits of 100,000 dollars. It has agreed to contribute 10 percent of its after-tax profits to the Red Cross Relief Fund. It must pay a provincial tax of 5 percent of its ...
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1answer
52 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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0answers
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 ...
4
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2answers
105 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 ...
0
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1answer
46 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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0answers
26 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
20 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 ...
2
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0answers
17 views

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 ...
2
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0answers
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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0answers
29 views

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 ...
5
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1answer
67 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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0answers
225 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
72 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.
2
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0answers
35 views

Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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1answer
22 views

Independence of distribution

Let there be a random matrix defined as $\mathbf{H}_1 = X + \boldsymbol\nu$, where, $X$ is deterministic and $\boldsymbol\nu$ is Gaussian white noise. Now let there be another random matrix defined as ...
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5answers
197 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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0answers
31 views

radius of convergence of hypergeometric functions

Hypergeometric function of scalar arguments is defined as \begin{eqnarray} _aF_b\left(p_1,...,p_a;q_1,...,q_b;z\right) &=&\sum_{i=0}^{\infty} ...
4
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1answer
148 views

Why does this determinant have a continuous density at zero?

This question is a simplification of my previous question. I think this is easy, but I don't have a strong enough background in probability. Let $A$ be a random $n\times n$ real matrix that satisfies ...
2
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1answer
62 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
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1answer
159 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
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0answers
30 views

Projection and matrix norm

Suppose we are in the matrix space $\mathbf{R}^{n_1 \times n_2}$. Suppose, $R_{\Omega}$ is an operator, such that $R_{\Omega}(Z)$ chooses $m$ entries from $Z$ uniformly at random with replacement and ...
0
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1answer
71 views

Important topics in Matrix analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
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0answers
63 views

Plotting the pair correlation function for the zeta zeros /GUE

I am making a shameless request for instructions on how to plot this: from this page. I can see from here that normalizing the zeros is given by ...
2
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0answers
73 views

Canonical Form of Nilpotent Matrices

Given the matrix $$\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$ where $S$ is an $n \times n$ ...
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0answers
67 views

Bounding the spectral norm of a block random matrix

Suppose that zero-mean iid random matrices $A_1 ,A_2,\dotsc,A_n$ satisfy $$\mathbb{P}\left(\left\|A_i \right\|\geq t\right)\leq \phi\left(t\right),\tag{*}$$ for $t>0$, where ...
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0answers
29 views

Distrbution of Matrix times vector

Given the distribution of an $n\times n$ matrix $A$, how to find the distribution of $Y = AX$, where $Y$ and $X$ are $n\times 1$ vector and X is deterministic.
5
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1answer
88 views

Closed formula for mean

Suppose we have the i.i.d. random variables $X_{11}, X_{12},\ldots, X_{nn}$, such that each $X_{ij}$ has standard normal distribution $N(0,1)$, with mean $0$ and variance $1$. Given some integer ...
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3answers
53 views

Random matrices in coordinate independent way

How to generate a random matrix in a basis independent way (so that the random distribution does not change if the coordinates are rotated)? I am especially interested in generating random rotation ...
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1answer
74 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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0answers
27 views

Distance of random span to a vector

I've been batteling with the following problem: Assume we have a diagonal matrix $D \in \mathbb{R}^{l \times l}$, a vector $\beta \in \mathbb{R}^l$. Next we simulate a random matrix (Idea inspired by ...
2
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1answer
49 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
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0answers
50 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
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1answer
49 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...