Find the transformation that diagonalizes a Hermitian Matrix?

Question

How do find the transformations in general?

Answer 1

The n x n matrix A you are trying to diagonalize must have n linearly independent eigenvectors. Then if you make S the n x n matrix with those eigenvectors as its columns, then

D = S^1AS is a diagonal matrix with diagonal entry d(ij) equal to the eigenvalue corresponding to the jth eigenvector in the S matrix.