Given a square matrix , Schur decomposition creates

where is unitary, and is upper diagonal,

with being the eigenvalues of . Note that if is Hermitian, is a diagonal matrix (with no terms).

Intuition#

We'll show this is possible via induction. With , this clearly works. Then, assume the result holds for . For matrix , let and be any eigenvalue and corresponding unit eigenvector. Extend with to form an orthonormal basis, so then with , we have

Now, since , the result holds, and we can express it as

Decomposing it in the result above, we arrive at the decomposition for ,

with and .