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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 GramSchmidt 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.
 Fall '12
 BernhardBodmann
 Matrices

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