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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 x2 , < Au, u >= λmax , which shows that the supremum in
x=0 < Ax, x >
= sup < A(
>= sup < Au, u >
||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 .
Then y is an eigenvector of A iﬀ ∀ diﬀerentiable f : R → S with f (0) = y , dt |t=0 qA (f (t)) = 0.
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