3
votes
1answer
101 views

Random Matrix Theory and ESD

I need some help to understand what professor Terence Tao means in this part of "Topics in random matrix theory". I'm having a hard time to undertand this function ESD. How is possible for it to be ...
0
votes
0answers
50 views

probability density of a real symmetric random matrix

I learn from Steve Lalley's lecture notes that the joint probability density of the entries of a real symmetric random matrix is a function only of the eigenvalues of the matrix. Then, let $RR^T$ be ...
0
votes
0answers
59 views

Given an innovations sequence, calculate transformation matrix, A

Let $Y_1,Y_2,Y_3,X)^T$ be a zero mean random vector with correlation matrix, $$ \begin{pmatrix} 2 & 1 & 1 & 2 \\ 1 & 2 & 1 & 2 \\ 1 & 1 & 2 & 2 \\ 2 & 2 & 2 ...
0
votes
1answer
71 views

Law of Gaussian Vector and Matrix

I have a probability problem that I’m struggling with. Any help would be appreciated. I have no clue about how to solve this. Here is the problem: Let A be: \begin{pmatrix} 4 & 7 \\ 1 & -2 ...
4
votes
1answer
1k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
2
votes
1answer
87 views

Orthogonal transformation on random matrices

Let $A$ and $B$ be two random matrices such that $A=U^T B V^T$. Here $U$ and $V$ are deterministic and orthogonal matrices. The random matrices $A$ and $B$ have the same distribution?
3
votes
0answers
31 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
1
vote
1answer
286 views

Computing the expected value of a matrix?

This question is about finding a covariance matrix and I wasn't sure about the final step. Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
3
votes
1answer
139 views

Why does this covariance matrix have additional symmetry along the anti-diagonals?

In my self study of a statistics book, I came across a page that has confused me somewhat. I am already familiar with covariance matricies, (or maybe not!), and the author's explanation leaves me a ...
2
votes
1answer
149 views

Variance and variance, covariance matrix

I'm trying to read about Principal Component Analysis and i'm stuck in the identity $Var[\alpha_{1}^{'}X]=\alpha_{1}^{'}\Sigma\alpha_{1}$ where $\Sigma$ is the variance-covariance matrix of $X$, ...