Schur's Theorem states that every
n x n matrix (or
square matrix) is
similar to an
upper triangular matrix.
That is, for any square matrix A, we can say that A = UTU*, where U is a unitary matrix, U* is the transpose of U, with each entry replaced by its complex conjugate and T is an upper triangular matrix. Also, if the entries and eigenvalues of A are all real, then there is an orthogonal U (A = UTUT, where UT is the transpose of U).