Structure of elements of the unitary group

[FunkyDwarf]

Hey guys,

I'm having a massive brain freeze here trying to show that for any element g in the unitary group you can always represent it as s*some diagonal matrix*s^-1. The only requirement for an element to be unitary is that its hermitian conjugate is its inverse correct? Any hints/help would be appreciated!

Cheers
-G

 

[shaggymoods]

http://en.wikipedia.org/wiki/Unitary_matrix

Check out the listing of equivalent definitions and see if any spark some ideas about diagonal matrices.

 

[FunkyDwarf]

Ah i think i see what youre saying, its kind of a basis transformation thing? I sort of understand that (not in a way to provide a proper proof though) but is there a way to arrive at that relation just from the strict definition of a unitary matrix?

Cheers
-G

 

[vizart]

It inevitably follows from a theorem in Linear algebra that states every unitary matrix is diagonalizable. You can probably find it in some L(Alg) textbook or alternatively you can simply google that statement.