Page 290 - A Course in Linear Algebra with Applications

274 Chapter Eight: Eigenvectors and Eigenvalues

Exercises 8.1
1. Find all the eigenvectors and eigenvalues of the following
matrices:

«»• (i J D' (I! i •!)•

£
2. Prove that tr(j4 + ) = tr(A) + tv(B) and tr(cA) = c tr(A)
where A and 5 are nxn matrices and c is a scalar.
3. If yl and B are nxn matrices, show that AB and BA have
the same eigenvalues. [Hint: let c be an eigenvalue of AB and
prove that it is an eigenvalue of BA ].
4. Suppose that A is a square matrix with real entries and
real eigenvalues. Prove that every eigenvalue of A has an
associated real eigenvector.
5. If A is a real matrix with distinct eigenvalues, then A is
diagonalizable over R: true or false?
6. Let p(x) be the polynomial

n
n 1
n 2
(-l) (x n + a n. xx - + a n_ 2x - + • • • + ao).
Show that p(x) is the characteristic polynomial of the follow-
ing matrix (which is called the companion matrix of p(x)):

/0 0 ••• 0 - a 0 \
1 0 ••• 0 - a i
0 1 ••• 0 -a 2

\ 0 0 ••• 1 - a n _ i /

7. Find matrices which diagonalize the following: