Linear Algebra, Theory And Applications, 2012a

316 SELF ADJOINT OPERATORS
The reason this is so is that the infimum is taken over a smaller set. Therefore, the infimum
gets larger. Now (13.9) is no larger than
⎧
⎫
⎨ k∑
⎬
inf
⎩ λ k |(x, u j )| 2 : |x| =1, (x, u j )=0forj>k, and x ∈ Y
⎭ = λ k
j=1
because since {u 1 , ··· ,u n } is an orthonormal basis, |x| 2 = ∑ n
j=1 |(x, u j)| 2 . It follows since
{w 1 , ··· ,w k−1 } is arbitrary,
{
sup inf
w 1,··· ,w k−1
{
(Ax, x) :|x| =1,x∈{w 1 , ··· ,w k−1 } ⊥}} ≤ λ k . (13.10)
However, for each w 1 , ··· ,w k−1 , the infimum is achieved so you can replace the inf in the
above with min. In addition to this, it follows from Corollary 13.3.4 that there exists a set,
{w 1 , ··· ,w k−1 } for which
{
inf (Ax, x) :|x| =1,x∈{w 1 , ··· ,w k−1 } ⊥} = λ k .
Pick {w 1 , ··· ,w k−1 } = {u 1 , ··· ,u k−1 } . Therefore, the sup in (13.10) is achieved and equals
λ k and (13.8) follows.
The following corollary is immediate.
Corollary 13.3.8 Let A ∈L(X, X) be self adjoint where X is a finite dimensional Hilbert
space. Then for λ 1 ≤ λ 2 ≤···≤λ n the eigenvalues of A, there exist orthonormal vectors
{u 1 , ··· ,u n } for which
Au k = λ k u k .
Furthermore,
{ {
}}
(Ax, x)
λ k ≡ max min
w 1,··· ,w k−1 |x| 2 : x ≠0,x∈{w 1 , ··· ,w k−1 } ⊥ (13.11)
where if k =1, {w 1 , ··· ,w k−1 } ⊥ ≡ X.
Here is a version of this for which the roles of max and min are reversed.
Corollary 13.3.9 Let A ∈L(X, X) be self adjoint where X is a finite dimensional Hilbert
space. Then for λ 1 ≤ λ 2 ≤···≤λ n the eigenvalues of A, there exist orthonormal vectors
{u 1 , ··· ,u n } for which
Au k = λ k u k .
Furthermore,
{ {
}}
(Ax, x)
λ k ≡ min max
w 1,··· ,w n−k |x| 2 : x ≠0,x∈{w 1 , ··· ,w n−k } ⊥ (13.12)
where if k = n, {w 1 , ··· ,w n−k } ⊥ ≡ X.