Page 270 - Advanced Linear Algebra
P. 270
254 Advanced Linear Algebra
or equivalently,
~
Now, is a polynomial in and is a polynomial in and so this holds if
and only if ~ .
Exercises
B
1. Let ²<Á = ³ . If is surjective, find a formula for the right inverse of
in terms of . If is injective, find a formula for a left inverse of in terms
i
of . Hint : Consider i and i .
i
=
B
2. Let ²= ³ where is a complex vector space and let
i i
b ² and ³ ~ ~ c ² ³
Show that and are self-adjoint and that
~ b and i ~ c
What can you say about the uniqueness of these representations of and
?
i
3. Prove that all of the roots of the characteristic polynomial of a skew-
Hermitian matrix are pure imaginary.
4. Give an example of a normal operator that is neither self-adjoint nor
unitary.
5. Prove that if ) ) ) #~ i ) ²#³ for all # = , where is complex, then is
=
normal.
6. Let be a normal operator on a complex finite-dimensional inner product
space or a self-adjoint operator on a real finite-dimensional inner product
=
space.
a Show that i ~ ² ³ , for some polynomial ²%³ ´%µ .
)
d
i
b Show that for any ²= ³ , ~ implies ~ . In other
i
)
B
i
words, commutes with all operators that commute with .
7. Show that a linear operator on a finite-dimensional complex inner product
space is normal if and only if whenever is an invariant subspace under
:
=
, so is : .
8. Let be a finite-dimensional inner product space and let be a normal
=
operator on .
=
)
a Prove that if is idempotent, then it is also self-adjoint.
b Prove that if is nilpotent, then ~ .
)
c Prove that if ~ , then is idempotent.
)
9. Show that if is a normal operator on a finite-dimensional complex inner
product space, then the algebraic multiplicity is equal to the geometric
multiplicity for all eigenvalues of .
10. Show that two orthogonal projections and are orthogonal to each other
if and only if im²³ im² . ³
