Page 287 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 271

also found eigenvectors for A; these form a matrix

/(-i + v=i)/2 -(i + v=T)/2y
5

Then by the preceding theory we may be sure that

Triangularizable matrices
It has been seen that not every complex square matrix is
diagonalizable. Compensating for this failure is the fact such
a matrix is always similar to an upper triangular matrix; this
is a result with many applications.
Let A be a square matrix over a field F. Then A is said
to be triangularizable over F if there is an invertible matrix S
l
over F such that S~ AS = T is upper triangular. It will also
be convenient to say that S triangularizes A. Note that the
diagonal entries of the triangular matrix T will necessarily be
the eigenvalues of A. This is because of Example 8.1.2 and the
fact that similar matrices have the same eigenvalues. Thus a
necessary condition for A to be triangularizable is that it have
n eigenvalues in the field F. When F = C, this condition is
always satisfied, and this is the case in which we are interested.

Theorem 8.1.8
Every complex square matrix is triangularizable.
Proof
Let A denote a n n x n complex matrix. We show by induction
on n that A is triangularizable. Of course, if n = 1, then A is
already upper triangular: let n > 1. We shall use induction
on n and assume that the result is true for square matrices
with n — 1 rows.