Chapter 7

Question 1

adjoint (T*)

Answer 1

suppose T in L(V,W). The adjoint of T is the function T: W –> V s.t. ⟨Tv, w⟩ = ⟨v, Tw⟩ for every v in V and every w in W
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Question 2

self-adjoint

Answer 2

an operator T in L(V) where T = T*.

In other words, T in L(V) is self-adjoint iff ⟨Tv, w⟩ = ⟨v, Tw⟩ for all v, w in V

Question 3

normal

Answer 3

• an operator on an inner product space that commutes with its adjoint
• in other words, T in L(V) s.t. TT* = T*T

Question 4

Complex Spectral Theorem

Answer 4

Suppose F = C and T in L(V). Then the following are equivalent:

(a) T is normal
(b) V has an orthonormal basis consisting of eigenvectors of T
(c) T has a diagonal matrix w.r.t. some orthonormal basis of V

Question 5

Real Spectral Theorem

Answer 5

Suppose F = R and T in L(V). Then the following are equivalent:
(a) T is self-adjoint
(b) V has an orthonormal basis consisting of eigenvectors of T
(c) T has a diagonal matrix w.r.t. some orthonormal basis of V
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Question 6

positive operator

Answer 6

An operator T in L(V) s.t.
T is self-adjoint
and ⟨Tv, v⟩ ≥ 0 for all v in V
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Question 7

characterization of positive operators

Answer 7

Let T in L(V). Then the following are equivalent:
(a) T is positive
(b) T is self-adjoint and all the eigenvalues of T are nonnegative
(c) T has a positive square root
(d) T has a self-adjoint square root
(e) there exists an operator R in L(V) s.t. T = R*R
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Question 8

singular values

Answer 8

suppose T in L(V). The singular values of T are the eigenvalues of sqrt(TT), with each eigenvalue λ repeated dim E(λ, sqrt(TT)) times
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Question 9

Singular Value Decomposition

Answer 9

suppose T in L(V) has singular values s1,…,sn. Then there exist orthonormal bases e1,…,en and f1,…,fn of V s.t. Tv = s1⟨v, e1⟩f1 + … + sn⟨v, en⟩fn for every v in V
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Question 10

What is the process of Singular Value decomposition?

Answer 10

(1) B := At * A
(2) get the eigenvalues of B
(3) their square roots are the singular values; sigma is a diagonal matrix with singular values along the diagonal.
(4) get normalized eigenvectors corresponding to the eigenvalues of B (note they are orthonormal)
(5) Find U such that A * V = U * sigma
(6) return A = (U * sigma * V^{-1})

Question 11

5 properties of the adjoint

Answer 11

(a) (S + T)* = S* + T* for all S, T in L(V, W)
(b) (λT)* = ˜λ˜(conjugate)T* for all λ in F and T in L(V, W)
(c) (T) = T for all T in L(V, W)
(d) I* = I, where I is the identity operator on V
(e) (ST)* = TS for all T in L(V, W) and S in L(W, U) (here U is an inner product space over F)

Question 12

the matrix of T*

Answer 12

Let T in L(V, W).
Suppose e1,…,en is an orthonormal basis of V and f1,…,fm is an orthonormal basis of W.

Then M(T*, (f1,...,fm), (e1,...,en)) is the conjugate transpose of M(T, (e1,...,en), (f1,...,fm))
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Question 13

eigenvalues of self-adjoint operators

Answer 13

(1) every self-adjoint operator over a nontrivial vector space has eigenvalues.
(2) every eigenvalue of a self-adjoint operator is real

Question 14

eigenvalues of the adjoint of a normal operator

Answer 14

Suppose T in L(V) is normal
and v in V is an eigenvector of T with eigenvalue λ.

Then v is also an eigenvector of T* with eigenvalue ˜λ˜ (conjugate)

Question 15

orthogonal eigenvectors for normal operators

Answer 15

Suppose T in L(V) is normal. Then eigenvectors of T corresponding to distinct eigenvalues are orthogonal.