P roof we can always find one nonzero eigenvector for

P
ROOF
. We can always find one nonzero eigenvector for each eigenvalue of
By the preceding theorem, the set is linearly independent. Thus is
diagonalizable by the Diagonalization theorem.

230 5. THE EIGENVALUE PROBLEM
Caution: Just because the matrix has fewer than distinct eigenvalues, you
may not conclude that it is not diagonalizable.
An example that illustrates this caution is part (a) of Example 5.1.13.
5.2 Exercises
1. Given each matrix below, find a matrix such that is diagonal. Use this
to deduce a formula for
(a) (b) (c)
2. Determine if the following matrices are diagonalizable with a minimum of calcula-
tion.
(a) (b) (c)
3. For each of the following matrices find the characteristic polynomial and
evaluate (This means that the matrix replaces every occurrence of and the
constant term is replaced by )
(a) (b) (c)
4. Suppose that is an invertible matrix which is diagonalized by the matrix that is,
is a diagonal matrix. Use this information to find a diagonalization for
5. Adapt the proof of Theorem 5.2.11 to prove that if eigenvectors are
such that for any eigenvalue of , the subset of all these vectors belonging to is
linearly independent, then the vectors are linearly independent.
6. Suppose that the kill rate of Example 5.2.2 is viewed as a variable positive param-
eter. There is a value of the number for which the eigenvalues of the corresponding
matrix are equal.
(a) Find this value of and the corresponding eigenvalues by examining the character-
istic polynomial of the matrix.
(b) Use the available MAS (or CAS) to determine experimentally the long term behavior
of populations for the value of found in (a). Your choices of initial states should
include
7. The thirteenth century mathematician Leonardo Fibonacci discovered the sequence
of integers called the Fibonacci sequence . These numbers have a way
of turning up in many applications. They can be specified by the formulas

5.2. SIMILARITY AND DIAGONALIZATION 231
(a) Let and show that these equations are equivalent to the matrix
equations and where
(b) Use part (a) and the diagonalization theorem to find an explicit formula for the th
Fibonacci number.
8. Calculate second, third and fourth powers of the matrix
Based on these calculations, make a conjecture about the form of where is
any positive integer.
9. Show that any upper triangular matrix with constant diagonal is diagonalizable if
and only if it is already diagonal. Hint: What diagonal matrix would such a matrix be
similar to?
10. Let be a transition matrix of a Markov chain where is not the identity
matrix.
(a) Show that can be written in the form for suitable real
numbers
(b) Show that and are eigenvectors for
(c) Use (b) to diagonalize the matrix and obtain a formula for the powers of
11. Show that if is diagonalizable, then so is
12. Let square matrices of the same size with invertible.
(a) Show that
(b) Show that
(c) Use (a) and (b) to show that if is a polynomial
function, then
13. Prove the Cayley-Hamilton theorem for diagonalizable matrices; that is, show that
if is the characteristic polynomial of the diagonalizable matrix then satisfies
its characteristic equation, that is, Hint: You may find Exercise 12 and
Corollary 5.2.4 very helpful.