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:

Linear Algebra on Jice Lavocat

Decomposition in unitary matrices