Page 291 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 275

w (a a) = 0.) (J j 1 ) •

8. For which values of a and b is the matrix I , 1 diago-
nalizable over C?

9. Prove that a complex 2 x 2 matrix is not diagonalizable
a b
if and only if it is similar to a matrix of the form
0 a
where 6 ^ 0 .
10. Let A be a diagonalizable matrix and assume that S is
a matrix which diagonalizes A. Prove that a matrix T diago-
nalizes A if and only if it is of the form T = CS where C is a
matrix such that AC = CA.
11. If A is an invertible matrix with eigenvalues ci,..., c n ,
1
show that the eigenvalues oi A~ l are c^~ ..., c^ .
,
12. Let T : V —>• V be a linear operator on a complex n-
dimensional vector space V. Prove that there is a basis
{vi, ..., v n } of V such that T(VJ) is a linear combination of
, for i = 1,... ,n.
v i 5 ... v n
13. Let T : P n (R) —• P n(R-) be the linear operator corre-
sponding to differentiation. Show that all the eigenvalues of
T are zero. What are the eigenvectors?

14. Let ci,...,c n be the eigenvalues of a complex matrix A.
Prove that the eigenvalues of A m are Cj",..., c™ where m is
any positive integer. [Hint: A is triangularizable].
15. Prove that a square matrix and its transpose have the
same eigenvalues.