Schur's Unitary Triangulization Theorem: For every square matrix A there exists some unitary matrix $U$ such that $$U^*AU=T$$ where $T$ is an upper triangular matrix with diagonal elements equal to the eigenvalues of $A$.
What's the intuition behind this theorem?
Apparently, it's equivalent to: for every nxn matrix, there exists an orthonormal basis in which the matrix is upper triangular. - but I don't see how.
Please help me out here!
Also, the choice of $U$ doesn't seem unique, so what's the best way to compute $U$?