### Wigner and Marcenko Pastur Theorems

I. Wigner’s theorem DefinitionLet . The probability distribution on defined by is called the semi-circular distribution. We denote . Wigner’s Theorem– Let i.i.d. centered, complex valued and with . – Let i.i.d. centered, real valued with . – The ‘s and the ‘s () are independent. – Consider and the hermitian matrices defined

### Davis-Kahan Sin Theta Theorem

In many situations, there is a symmetric matrix of interest , but one only has a perturbed version of it . How is affected by ?The basic example is Principal Component Analysis (PCA). Let for some random vector , and let be the sample covariance matrix on independent copies of . Namely, we observe i.i.d.

### Slepian and Gordon Theorems with application

In this post, we present two important results of the field of Gaussian Processes: the Slepian and Gordon Theorems. For both of them, we illustrate their power giving specific applications to random matrix theory. More precisely, we derive bounds on the expected operator norm of gaussian matrices. 1. Slepian’s Theorem and application Theorem (Slepian Fernique

### Compressed sensing : A brief introduction

Compressed sensing belongs to the large field of inverse problems. A typical example of such problems consists in determining the signal that produces the measurement vector through the linear transformation , namely . Of course, without additional assumption the problem is ill-posed if the rank of the matrix is strictly smaller than , meaning that

### Perron Frobenius Theorem

Perron Frobenius Theorem is a well-known algebra result that finds applications in a large span of fields of Mathematics. One can mention Markov chains, population growth (with the Leslie matrix model) or the famous PageRank algorithm. In this post we prove the Perron-Frobenius Theorem for stochastic matrices. Perron Frobenius Theorem Let us consider some integer