Suppose A is a unitary matrix.
(a) Show that there exists an orthonormal basis B of eigenvectors for A.
(b) Let P be the associated change-of-basis matrix. Explain how to alter B such that P lies in SU(n).
(a) We will prove first a lemma:
Lemma: If A is a unitary and x,y are two distinct eigenvalues and v,w 2 eigenvectors corresponding to them, then x is orthogonal on y, which is the same as saying (x,y)=0.
((x,y) means the inner product of x and y, also denoted sometimes by (x|y) or by x.y)
Proof of the lemma:
First, because x and y are eigenvalues or a unitary matrix, they have absolute value 1. Also, U unitary implies:
(Uv,Uw)=(v,w). On the other hand, because v,w are eigenvectors, we have: Uv=x*v and Uw=y*w so if we use this in the previous equality we get:
(xv,yw)=(v,w) so because x,y ...
This is a proof regarding a unitary matrix. The associated change-of basis matrix is examined.