Maths on Jice Lavocat

Tout sur Hadamard

Mon, 20 Jul 2009 00:00:00 +0000

Qui est donc ce fameux Hadamard dont tout le monde parle en informatique quantique? Jacques Hadamard est un mathématicien français (1865-1963) à qui l’on doit les célèbres matrices (encore un normalien).

Je vous conseille la biographie suivante sur Jacques Hadamard : Jacques Hadamard un mathématicien universel

Ainsi que ce site regroupant les propriétés des-dites matrices : http://chaos.if.uj.edu.pl/~karol/hadamard/index.php

On Matrix decomposition

Wed, 08 Jul 2009 00:00:00 +0000

In order to complete the previous post on unitary matrix decomposition (sum) , I give here two interesting links about other known matrix decomposition :

Decomposition in unitary matrices

Tue, 07 Jul 2009 00:00:00 +0000

Conjecture : Any complex or real matrix is the sum of two unitary matrices.

Proof (ideas) :

We know that every complex matrix A could be diagonalized using two unitary matrices U and V : $$ A = UDV^{*} $$ . The matrix D has positive elements : D=diag(d1,…d2) with $$d_1\geq d_2 \geq …\geq d_n \geq 0$$.

A basic result is the following : every diagonal matrix could be diagonalized with n unitary matrix. Indeed, you just have to choose the good coefficient and use the set of matrix {$$ E_i $$} where $$E_i$$ is a diagonal with -1 everywhere, except at position i where there is a +1: