Page 207 - Advanced Linear Algebra

Eigenvalues and Eigenvectors 191

~ ´² c ³ ²# ³µ

Á

~² c b ³´² c ³ ²# ³µ

Á

~² c ³ b ²# Á ³ b ² c ³ ²# ³

Á

~ b b
For ~ c , a similar computation, using the fact that
Á
²c ³ b ²# ³ ~ ²c ³ ²# ³ ~

Á
Á
Á
gives
³ ~ ² c c
Á Á
Thus, for this basis, the complexity is more or less spread out evenly, and the
matrix of with respect to Á is the Á d Á matrix
9O
ºº# »»
Á
v Ä Ä y
x Æ Å {
x {
@ ²Á ³ ~ x Á Æ Æ Å {
x {
Å ÆÆÆ
w Ä z
which is called a Jordan block associated with the scalar . Note that a Jordan

block has 's on the main diagonal, 's on the subdiagonal and 's elsewhere.

Let us refer to the basis
9 9 ~ Á
as a Jordan basis for .

Theorem 8.6 ( The Jordan canonical form) Suppose that the minimal
polynomial of ²= ³ splits over the base field , that is,
-

B
²%³ ~ ²% c ³ IJ% c ³

where - .
)
1 The matrix of with respect to a Jordan basis is
9

diag Á @ Á Á ² ³ÁÃÁ @ Á Á ³ÁÃÁ @ Á Á ² ³ @ ² Á ³ÁÃÁ
²

where the polynomials ²% c ³ Á are the elementary divisors of . This
block diagonal matrix is said to be in Jordan canonical form and is called
the Jordan canonical form of .
)
2 If - is algebraically closed, then up to order of the block diagonal
matrices, the set of matrices in Jordan canonical form constitutes a set of
canonical forms for similarity.
Proof. For part 2), the companion matrix and corresponding Jordan block are
similar: