# Unitary Matrix Proof

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).

#### Solution Preview

...are scalars they go outside the inner product, second one conjugate:

x*conj(y)*(v,w)=(v,w) (I denoted by conj(y) the conjugate of y, usually denoted by y bar).

Now if v and w wouldn't be orthogonal (reasoning by contradiction) this is the same as saying that (v,w) is not zero so the we can divide by it in the previous equality so we get:

x*conj(y)=1 which means x/y=1 because y*conj(y)=the square of the absolute value of y which is 1, so conj(y)=1/y.

So x/y=1 means x=y, contradiction with the hypothesis that x and y were different. This contradiction shows that (v,w) has to be zero which ...