Page 201 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
190
19.
operator on V has a cube root.
20. Prove that if V is a complex vector space, then every invertible
Suppose T ∈L(V) is invertible. Prove that there exists a polyno-
mial p ∈P(F) such that T −1 = p(T).
3
21. Give an example of an operator on C whose minimal polynomial
2
equals z .
4
22. Give an example of an operator on C whose minimal polynomial
2
equals z(z − 1) .
For complex vector 23. Suppose V is a complex vector space and T ∈L(V). Prove that
spaces, this exercise V has a basis consisting of eigenvectors of T if and only if the
adds another minimal polynomial of T has no repeated roots.
equivalence to the list
given by 5.21. 24. Suppose V is an inner-product space. Prove that if T ∈L(V) is
normal, then the minimal polynomial of T has no repeated roots.
25. Suppose T ∈L(V) and v ∈ V. Let p be the monic polynomial of
smallest degree such that
p(T)v = 0.

Prove that p divides the minimal polynomial of T.
4
26. Give an example of an operator on C whose characteristic and
2
minimal polynomials both equal z(z − 1) (z − 3).
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27. Give an example of an operator on C whose characteristic poly-
2
nomial equals z(z − 1) (z − 3) and whose minimal polynomial
equals z(z − 1)(z − 3).
This exercise shows 28. Suppose a 0 ,...,a n−1 ∈ C. Find the minimal and characteristic
n
that every monic polynomials of the operator on C whose matrix (with respect to
polynomial is the the standard basis) is
characteristic  
0 −a 0
polynomial of some  
 1 0 −a 1 
operator.  
 . . 
 1 . −a 2 
 
 . .  .
 . . 
 . . 
 
 0 
 −a n−2 
1 −a n−1