Page 280 - A Course in Linear Algebra with Applications

2 6 4 Chapter Eight: Eigenvectors and Eigenvalues

The other coefficients in the characteristic polynomial are
not so easy to describe, but they are in fact expressible as
subdeterminants of det(i4). For example, take the case of
n 2 n 2
x ~ . Now terms in x ~ arise in two ways: from the product
(an — x) • • • (a nn — x) or from products like

-ai 2 a 2 i(a 3 3 - x) • • • (a nn - x).

n 2
So a typical contribution to the coefficient of x ~ is

n 2 an a i 2
(-l) - (ana 2 2 - a 12a 2i) = (-1)
0-21 0-22
From this it is clear that the term of degree n — 2 in p(x) is
n 2
just (—x) ~ times the sum of all the 2 x 2 determinants of
the form
an O'ij
a a
ji jj
where i < .
j
In general one can prove by similar considerations that
the following is true.
Theorem 8.1.2
The characteristic polynomial of the n x n matrix A equals

n
n
J2di(-x) -*
i=0

where di is the sum of all the i x i subdeterminants of det(A)
whose principal diagonals are part of the principal diagonal of
A.

Now assume that the matrix A has complex entries. Let
ci, c 2 ,..., c n be the eigenvalues of A. These are the n roots of
the characteristic polynomial p(x). Therefore, allowing for the