N 1 n suppose schurs theorem holds for all n 1 n

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Unformatted text preview: uare matrix is unitarily equivalent to an upper triangular matrix. Proof. The proof follows by induction on the dimension, n: (n = 1) : Done. (n − 1 ⇒ n) : Suppose Schur’s theorem holds for all (n − 1) × (n − 1) matrices, and let A ∈ Mn , with eigenvalues λ1 , λ2 , ..., λn . Choose an eigenvalue x corresponding to λ1 and, WLOG, suppose ||x|| = 1. Now extend {x} to a basis for Cn , and then apply the Gram-Schmidt to get an orthonormal basis: {x, z2 , z3 , ..., zn }. Next form the matrix U1 defined by: ￿ ￿ U1 = x z2 z3 · · · zn Then U1 is unitary, since its columns are orthonormal. (Refer to the characterizations of unitary matrices from last ime.) Noting that we chose x so that Ax = λ1 x, it follows that: ∗ ∗ U1 AU1 = U1 ￿ λ1 x ∗ · · · ∗...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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