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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).

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