MAT067 University of California, Davis Winter 2007
Homework Set 6: Exercises on Eigenvalues
Directions: Please work on all exercises and hand in your solutions to Problems 6 and 7 at
the beginning of lecture on February 16, 2006. (Because of the midterm there is only one
set of homeworks this week!)
As usual, we are using F to denote either R or C, and F[z] denotes the set of polynomials
with coefficients over F.
1. Let V be a finite-dimensional vector space over F, and let S, T ∈ L(V ) be linear
operators on V with S invertible. Given any polynomial p(z) ∈ F[z], prove that
p(S ◦ T ◦ S −1 ) = S ◦ p(T ) ◦ S −1 .
2. Let V be a finite-dimensional vector space over C, T ∈ L(V ) be a linear operator on
V , and p(z) ∈ C[z] be a polynomial. Prove that λ ∈ C is an eigenvalue of the linear
operator p(T ) ∈ L(V ) if and only if T has an eigenvalue µ ∈ C such that p(µ) = λ.
3. Let V be a finite-dimensional vector space over C with T ∈ L(V ) a linear operator
on V . Prove that, for each k = 1, . . . , dim(V ), there is an invariant subspace Uk of V
under T such that dim(Uk ) = k.
4. Prove or give a counterexample to the following claim:
Claim. Let V be a finite-dimensional vector space over F, and let T ∈ L(V ) be a linear
operator on V . If the matrix for T with respect to some basis on V has all zeros on
the diagonal, then T is not invertible.
5. Prove or give a counterexample to the following claim:
Claim. Let V be a finite-dimensional vector space over F, and let T ∈ L(V ) be a linear
operator on V . If the matrix for T with respect to some basis on V has all non-zero
elements on the diagonal, then T is invertible.
6. Let V be a finite-dimensional vector space over F, and let S, T ∈ L(V ) be linear
operators on V . Suppose that T has dim(V ) distinct eigenvalues and that, given any
eigenvector v for T associated to some eigenvalue λ, v is also an eigenvector for S
associated to some (possibly distinct) eigenvalue µ. Prove that T ◦ S = S ◦ T .
7. Let V be a finite-dimensional vector space over F, and suppose that the linear operator
P ∈ L(V ) has the property that P 2 = P . Prove that V = null(P ) ⊕ range(P ).
