Page 279 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 263

Properties of the characteristic polynomial
Now let us see what can be said in general about the
o
characteristic polynomial fannxn matrix A. Let p(x) denote
this polynomial; thus

an — x a\2
0-21 0-22 - X
p(x) =

0"nl 0-n2 Q>nn 2-

At this point we need to recall the definition of a determinant
as an alternating sum of terms, each term being a product of
entries, one from each row and column. The term of p(x) with
highest degree in x arises from the product

(an - x)--- (a nn - x)

n
and is clearly (—x) . The terms of degree n — 1 are also easy
to locate since they arise from the same product. Thus the
n x
coefficient of x ~ is

n 1
( - l ) - ( a n + --- + a nn )

and the sum of the diagonal entries of A is seen to have sig-
nificance; it is given a special name, the trace of A,

tr(A) = an + a 22 H h a nn.

-1
The term in p(x) of degree n — 1 is therefore tr(^4) (—a;)" .
The constant term in p(x) may be found by simply
putting x = 0 in p(x) = det(A — xl n), thereby leaving det(A).
Our knowledge of p(x) so far is summarized in the formula

p{x) = (-x) n + r ^ X - a : ) " - 1 + • • • + det(A).
t