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Lecture #13 Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from October 9, 2012 taken by Charles Mills Last Time (10/6/12) Suppose A M n satisfies A = A , then given A , we can define a map q A : C n R by q A ( x ) = < Ax, x > . It is clear that if we know the matrix A , we know q A and vice versa. This section is about obtaining information about the eigenvalues of A from q A . 4 Variational characterization of eigenvalues 4.1 Rayleigh-Ritz We introduce some notation. In the following section, we always order the eigenvalues of a Hermitian A such that λ min = λ 1 λ 2 · · · λ n = λ max , multiplicity counted. 4.1.1 Theorem. (Rayleigh-Ritz) Let A C n satsify A = A , then: 1. x C n , λ min || x || 2 < Ax, x > λ max || x || 2 . 2. λ max = max || x || =1 < Ax, x > = max x =0 <Ax,x> || x || 2 λ min = min || x || =1 < Ax, x > = min x =0 <Ax,x> || x || 2 Proof. Since A = A , A is diagonalizable by a unitary matrix U . Let U be such that UAU = D D is diagonal with d ii = λ i U U = I U = [ U 1 , U 2 , . . . , U n ] where AU i = λ i U i 1. Given such an A, U, and D
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