This preview shows page 1. Sign up to view the full content.
Unformatted text preview: or λmax , by choosing or Ax = λmax x and thus obtaining < Ax, x >=
λmax x2 , < Ax, x >= λmax x2 , < 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 x2
x=0
u=1 is attained and equals
max < Ax, x >= max x=1 x=0 < Ax, x >
= λmax .
x2 As a direct consequence of the RayleighRitz 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. RayleighRitz 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...
View
Full
Document
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
 Math, Matrices

Click to edit the document details