X2 as a direct consequence of the rayleigh ritz

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Unformatted text preview: or λmax , by choosing or Ax = λmax x and thus obtaining < Ax, x >= λmax ||x||2 , < Ax, x >= λmax ￿x￿2 , < Au, u >= λmax , which shows that the supremum in sup x￿=0 < Ax, x > x x = sup < A( ), >= sup < Au, u > 2 2 ||x|| ||x|| ||x||2 x￿=0 ||u||=1 is attained and equals max < Ax, x >= max ||x=1|| x￿=0 < Ax, x > = λmax . ||x||2 As a direct consequence of the Rayleigh-Ritz theorem we have the following corollary. 2 4.1.2 Corollary. If A = A∗ , and < Ax, x >= α for some x ∈ Cn with ||x|| = 1, then λmin ≤ α ≤ λmax . 4.1.3 Question. Rayleigh-Ritz on gives us information about λmin and λmax , but can we use < Ax, x > to obtain any information about λ2 . . . λn−1 ? It turns out that qA does indeed encode information about λ2 . . . λn−1 which we demonstrate in the next section Other eigenvalues of A 4.1.4 Theorem. Let A ∈ Mn satisfy A = A∗ . Let S be the unit sphere in Cn . Take y ∈ S . d Then y is an eigenvector of A iff ∀ differentiable f : R ￿→ S with f (0) = y , dt |t=0 qA (f (t)) = 0. Befor...
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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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