Fundamentals of Matrix Algebra, 2011a

Chapter 4
Eigenvalues and Eigenvectors
.
Ḋefinion 28
Characterisc Polynomial
.
Let A be an n × n matrix. The characterisc polynomial of A
is the n th degree polynomial p(λ) = det (A − λI).
Our definion just states what the characterisc polynomial is. We know from our
work so far why we care: the roots of the characterisc polynomial of an n × n matrix
A are the eigenvalues of A.
In Examples 84 and 85, we found eigenvalues and eigenvectors, respecvely, of
a given matrix. That is, given a matrix A, we found values λ and vectors ⃗x such that
A⃗x = λ⃗x. The steps that follow outline the general procedure for finding eigenvalues
and eigenvectors; we’ll follow this up with some examples.
.
Finding Eigenvalues and Eigenvectors
. Key Idea 14
Let A be an n × n matrix.
1. To find the eigenvalues.
of A, compute p(λ), the characterisc
polynomial of A, set it equal to 0, then solve
for λ.
2. To find the eigenvectors of A, for each eigenvalue
solve the homogeneous system (A − λI)⃗x = ⃗0.
. Example 86 .Find the eigenvalues of A, and for each eigenvalue, find an eigenvector
where
[ ] −3 15
A =
.
3 9
S
equal to 0.
To find the eigenvalues, we must compute det (A − λI) and set it
det (A − λI) =
∣
−3 − λ 15
3 9 − λ
∣
= (−3 − λ)(9 − λ) − 45
= λ 2 − 6λ − 27 − 45
= λ 2 − 6λ − 72
= (λ − 12)(λ + 6)
Therefore, det (A − λI) = 0 when λ = −6 and 12; these are our eigenvalues. (We
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