370 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Then from Corollary 13.13.5,

λ k = inf{(Ax,x) : |x|= 1,x ∈ {u1, · · · ,uk−1}⊥

}≤

supw1,··· ,wk−1

{inf{(Ax,x) : |x|= 1,x ∈ {w1, · · · ,wk−1}⊥

}}≤ λ k

Hence these are all equal and this proves the theorem. ■The following corollary is immediate.

Corollary 13.13.9 Let A ∈L (X ,X) be self adjoint where X is a finite dimensional innerproduct space. Then for λ 1 ≤ λ 2 ≤ ·· · ≤ λ n the eigenvalues of A, there exist orthonormalvectors {u1, · · · ,un} for which

Auk = λ kuk.

Furthermore,

λ k ≡ maxw1,··· ,wk−1

{min

{(Ax,x)

|x|2: x ̸= 0,x ∈ {w1, · · · ,wk−1}⊥

}}(13.29)

where if k = 1,{w1, · · · ,wk−1}⊥ ≡ X .

Here is a version of this for which the roles of max and min are reversed.

Corollary 13.13.10 Let A ∈L (X ,X) be self adjoint where X is a finite dimensional innerproduct space. Then for λ 1 ≤ λ 2 ≤ ·· · ≤ λ n the eigenvalues of A, there exist orthonormalvectors {u1, · · · ,un} for which

Auk = λ kuk.

Furthermore,

λ k ≡ minw1,··· ,wn−k

{max

{(Ax,x)

|x|2: x ̸= 0,x ∈ {w1, · · · ,wn−k}⊥

}}(13.30)

where if k = n,{w1, · · · ,wn−k}⊥ ≡ X .

13.14 Positive and Negative Linear TransformationsThe notion of a positive definite or negative definite linear transformation is very importantin many applications. In particular it is used in versions of the second derivative test forfunctions of many variables. Here the main interest is the case of a linear transformationwhich is an n× n matrix but the theorem is stated and proved using a more general nota-tion because all these issues discussed here have interesting generalizations to functionalanalysis.

Definition 13.14.1 A self adjoint A ∈ L (X ,X) , is positive definite if whenever x ̸= 0,(Ax,x)> 0 and A is negative definite if for all x ̸= 0, (Ax,x)< 0. A is positive semidef-inite or just nonnegative for short if for all x, (Ax,x) ≥ 0. A is negative semidefinite ornonpositive for short if for all x, (Ax,x)≤ 0.