Page 171 - Linear Algebra Done Right

Exercises
159
3
8.
such that T(1, 2, 3) = (0, 0, 0) and T(2, 5, 7) = (2, 5, 7).
Prove that a normal operator on a complex inner-product space
9. Prove that there does not exist a self-adjoint operator T ∈L(R ) Exercise 9 strengthens
is self-adjoint if and only if all its eigenvalues are real. the analogy (for normal
operators) between
10. Suppose V is a complex inner-product space and T ∈L(V) is a self-adjoint operators
9
8
normal operator such that T = T . Prove that T is self-adjoint and real numbers.
2
and T = T.
11. Suppose V is a complex inner-product space. Prove that every
normal operator on V has a square root. (An operator S ∈L(V)
2
is called a square root of T ∈L(V) if S = T.)
12. Give an example of a real inner-product space V and T ∈L(V) This exercise shows
2
2
and real numbers α, β with α < 4β such that T + αT + βI is that the hypothesis
not invertible. that T is self-adjoint is
needed in 7.11, even
13. Prove or give a counterexample: every self-adjoint operator on for real vector spaces.
V has a cube root. (An operator S ∈L(V) is called a cube root
3
of T ∈L(V) if S = T.)
14. Suppose T ∈L(V) is self-adjoint, λ ∈ F, and > 0. Prove that if
there exists v ∈ V such that v = 1 and

Tv − λv < ,

then T has an eigenvalue λ such that |λ − λ | < .

15. Suppose U is a finite-dimensional real vector space and T ∈
L(U). Prove that U has a basis consisting of eigenvectors of T if
and only if there is an inner product on U that makes T into a
self-adjoint operator.
16. Give an example of an operator T on an inner product space such This exercise shows
that T has an invariant subspace whose orthogonal complement that 7.18 can fail
is not invariant under T. without the hypothesis
that T is normal.
17. Prove that the sum of any two positive operators on V is positive.
k
18. Prove that if T ∈L(V) is positive, then so is T for every positive
integer k.