Page 271 - Advanced Linear Algebra

Structure Theory for Normal Operators 255

11. Let be a normal operator and let be any operator on = . If the
eigenspaces of are -invariant, show that and commute.

12. Prove that if and are normal operators on a finite-dimensional complex

inner product space and if ~ for some operator then ~ i i .
13. Prove that if two normal d complex matrices are similar, then they are
unitarily similar, that is, similar via a unitary matrix.
14. If is a unitary operator on a complex inner product space, show that there

exists a self-adjoint operator for which ~ .
15. Show that a positive operator has a unique positive square root.
16. Prove that if has a square root, that is, if ~ , for some positive
operator , then is positive.

17. Prove that if  (that is, c is positive) and if is a positive operator

that commutes with both and then . 

18. Using the 89 factorization, prove the following result, known as the
B
Cholsky decomposition. An invertible linear operator ²= ³ is positive

if and only if it has the form ~ i where is upper triangularizable.
Moreover, can be chosen with positive eigenvalues, in which case the

factorization is unique.
19. Does every self-adjoint operator on a finite-dimensional real inner product
space have a square root?
20. Let be a linear operator on d and let be the eigenvalues of ,
ÁÃÁ

each one written a number of times equal to its algebraic multiplicity. Show
that
(( i ² tr ³

where is the trace. Show also that equality holds if and only if is
tr

normal.
=
21. If B ²= ³ where is a real inner product space, show that the Hilbert
i

space adjoint satisfies ²³ ~ ² d ³ i d .