Page 277 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 261

Theorem 8.1.1
Let A be an n x n complex matrix.
(i) The eigenvalues of A are precisely the n roots of the
characteristic polynomial &et(A — xl n);
(ii) the eigenvectors of A associated with an eigenvalue c
are the non-zero vectors in the null space of the matrix
A-cI n.
Thus in Example 8.1.1 the characteristic polynomial of
the matrix is

2-x - 1 = x 2 - 6x + 10.
2 4-x

The eigenvalues are the roots of the characteristic equation
2
x — 6x + 10 = 0, that is, c\ = 3 + \f—T and c^ — 3 — \J—1;
the eigenspaces of c\ and c^ are generated by the vectors

( _ l + v /3T ) / 2 -(l + V=l)/2
and
1
respectively.

Example 8.1.2
Find the eigenvalues of the upper triangular matrix

(a\\-x ai2 ai3 « l n \
0 a 22 - x a 23 0-2n

\ 0 0 0 a nn — x /

The characteristic polynomial of this matrix is

i
an - x a 12 a 13 a n
0 a 22 - x a 23 0>2n
0 0 0 Ojn.n. ^