Page 276 - A Course in Linear Algebra with Applications

260 Chapter Eight: Eigenvectors and Eigenvalues

Conversely, if the scalar c satisfies this equation, there will
be a non-zero solution of the system and c will be an eigen-
value. These considerations already make it clear that the
determinant

-x • • •
a n a 12 a ln
«2i a 22~ x • • •
det(A - xl n) = a 2n
0"nl Q"n2 ' ' ' &nn X
must play an important role. This is a polynomial of de-
gree n in x which is called the characteristic polynomial of
A. The equation obtained by setting the characteristic poly-
nomial equal to zero is the characteristic equation. Thus the
eigenvalues of A are the roots of the characteristic equation
(or characteristic polynomial) which lie in the field F.
At this point it is necessary to point out that A may
well have no eigenvalues in F. For example, the characteristic
polynomial of the real matrix

is x 2 + 1, which has no real roots, so the matrix has no eigen-
values in R.
However, if A is a complex nxn matrix, its characteristic
equation will have n complex roots, some of which may be
equal. The reason for this is a well-known result known as
The Fundamental Theorem of Algebra; it asserts that every
polynomial / of positive degree n with complex coefficients can
be expressed as a product of n linear factors; thus the equation
f(x) = 0 has exactly n roots in C. Because of this we can be
sure that complex matrices always have all their eigenvalues
and eigenvectors in C. It is this case that principally concerns
us here.
Let us sum up our conclusions about the eigenvalues of
complex matrices so far.