Page 371 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 355

Example 9.4.6
The n x n upper triangular matrix

( c 1 0 •• • 0 °\
0 c 1 •• • 0 0
0 0 c • •• 0 0
A
0 0 0 •• c 1
Ko 0 0 •• • 0 cJ

n
has minimum polynomial (x —c) ; this is because (A — cl) n =
n 1
0, but (A — cl) ~ / 0. Hence A is diagonalizable if and only
if n = 1. Notice that the characteristic polynomial of A equals
n
(c-x) .
Jordan normal form
We come now to the definition of the Jordan normal form
of a square complex matrix. The basic components of this are
certain complex matrices called Jordan blocks, of the type
considered in Example 9.4.6. In general a n n x n Jordan block
is a matrix of the form

0 0\
I c ± u
0 c 1 0 0
J 0 0 c 0 0

0 0 0 c 1
V n n n 0 c)

for some scalar c. Thus J is an upper triangular n x n ma-
trix with constant diagonal entries, a superdiagonal of l's, and
zeros elsewhere. By Example 9.4.6 the minimum and charac-
teristic polynomials of J are (x — c) n and (c — x) n respectively.
We must now take note of the essential property of the
matrix J. Let E±,..., E n be the vectors of the standard basis
n
of C . Then matrix multiplication shows that JE\ = cEi,
and JEi = cEi + -E^-i where 1 < i < n.