Find an orthogonal or unitary matrix P and a diagonal matrix D such that

a)

b)

My solution so far:

From an example in the book, I know that I have to find the eigenvalues of A, then find a basis for the eigenspaces. Use the Gram-Schmidt process to obtain the orthogonal basis, take the union of these bases, then they will make up the entries of P.

Now, for a), I have eigenvalues i and -i, but I have difficulty trying to find the eigenspaces, I have

But I forgot how to find a basis of that, I know I have done this before, but I just can't do it again now.

Thanks.