GLOBALLY ANALYTIC TRIANGULARIZATION OF A MATRIX FUNCTION

A well-known fundamental theorem of Schur guarantees that every constant matrix A is unitarily similar to a triangular matrix. If A(t) is a matrix-valued function analytic on [a,b] with real eigenvalues, it is shown in this paper via a constructive algorithm that there exists a unitary matrix U(t) analytic on [a,b] such that U*(t)A(t)U(t) is triangular. In the process of doing so, we also show that the Gram-Schmidt orthogonalization process preserves global analyticity. A similar algorithm which preserves periodicity works also for a periodic analytic matrix.