Answer to Question #154750 in Linear Algebra for deviika

Question

prove that the eigen values of unitary matrix are unit modulus

Answer 1

Proof:

Let U be a unitary matrix and let E be the eigen value with corresponding eigen vector v.

By definition of unitary operator,

$($v,v$)$= $($ Uv, Uv$)$

But

Uv=Ev

so

$($v,v$)$= $($ Ev, Ev$)$

=E$\widetilde {E}$$($v,v$)$

= $\vert E \vert$²$($v,v$)$

Since

  $($v,v$)$$\not =$0

we can have  $\vert E \vert$²=1

That is  $\vert E \vert$=1

Hence, the eigen values of unitary matrix are unit modulus.