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* = *UTU*^{T}, where *U*^{T} is the transpose of U).