Page 289 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 273

Q-lAQ— ( C A ^ \ - ( C A ^
b Ab
~ \0 S^A S ) ~ \0 B 2
X 2
This matrix is clearly upper triangular, so the theorem is
proved.

The proof of the theorem provides a method for triangu-
lar izing a matrix.
Example 8.1.6

Triangularize the matrix A -
- 1 3 /
The characteristic polynomial of A is x 2 — 4x + 4, so both
eigenvalues equal 2. Solving (A — 2I 2)X = 0, we find that

all the eigenvectors of A are scalar multiples of X\ =

Hence A is not diagonalizable by 8.1.6.
Let T be the linear operator on C 2 arising from left mul-
tiplication by A. Adjoin a vector to X 2 to X\ to get a basis
2
= X 2} of C , say = (J J. Denote by the
B 2 {X u X 2 B x
2
standard basis of C . Then the change of basis B\ —> B 2 is
described by the matrix Si = ( ). Therefore by 6.2.6

the matrix A which represents T with respect to the basis B 2
is

Hence S = S^ 1 = I j triangularizes A.