Page 281 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 265

fact that the term of p(x) with highest degree has coefficient
n
(—l) , one has

p(x) = (ci - x)(c 2 -x)---(c n- x).

The constant term in this product is evidently just c\Ci... c n,
n l n-1
while the term in x ~ has coefficient (—l) (ci + • • • + c n ).
On the other hand, we previously found these to be det(A) and
n 1
(—l) ~ tx{A) respectively. Thus we arrive at two important
relations between the eigenvalues and the entries of A.

Corollary 8.1.3
// A is any complex square matrix, the product of the eigenval-
ues equals the determinant of A and the sum of the eigenvalues
equals the trace of A

Recall from Chapter Six that matrices A and B are said
to be similar if there is an invertible matrix S such that B =
X
SAS~ . The next result indicates that similar matrices have
much in common, and really deserve their name.

Theorem 8.1.4
Similar matrices have the same characteristic polynomial and
hence they have the same eigenvalues, trace and determinant.

Proof
The characteristic polynomial of B — SAS^ 1 is

1
det^SAS' 1 - xl) =det(S(A - x^S' )
= det(S) det(A - xl) det(5)" 1
= det{A-xI).

Here we have used two fundamental properties of determi-
nants established in Chapter Three, namely 3.3.3 and 3.3.5.
The statements about trace and determinant now follow from
8.1.3.