Page 172 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
160
19.
Suppose that T is a positive operator on V. Prove that T is in-
vertible if and only if
Tv, v > 0
for every v ∈ V \{0}.
2
20. Prove or disprove: the identity operator on F has infinitely many
self-adjoint square roots.
21. Prove or give a counterexample: if S ∈L(V) and there exists
an orthonormal basis (e 1 ,...,e n ) of V such that Se j = 1 for
each e j , then S is an isometry.

3
22. Prove that if S ∈L(R ) is an isometry, then there exists a nonzero
3
2
vector x ∈ R such that S x = x.
3
23. Define T ∈L(F ) by
T(z 1 ,z 2 ,z 3 ) = (z 3 , 2z 1 , 3z 2 ).

√
3
Find (explicitly) an isometry S ∈L(F ) such that T = S T T.
∗
Exercise 24 shows that 24. Suppose T ∈L(V), S ∈L(V) is an isometry, and R ∈L(V) is a
√
if we write T as the positive operator such that T = SR. Prove that R = T T.
∗
product of an isometry
and a positive operator 25. Suppose T ∈L(V). Prove that T is invertible if and only if there
√
∗
(as in the polar exists a unique isometry S ∈L(V) such that T = S T T.
decomposition), then
26. Prove that if T ∈L(V) is self-adjoint, then the singular values
the positive operator
√ of T equal the absolute values of the eigenvalues of T (repeated
must equal T T.
∗
appropriately).
27. Prove or give a counterexample: if T ∈L(V), then the singular
2
values of T equal the squares of the singular values of T.
28. Suppose T ∈L(V). Prove that T is invertible if and only if 0 is
not a singular value of T.

29. Suppose T ∈L(V). Prove that dim range T equals the number of
nonzero singular values of T.

30. Suppose S ∈L(V). Prove that S is an isometry if and only if all
the singular values of S equal 1.