8.7. JORDAN CANONICAL FORM
411
Let
A
be an
n
–by–
n
matrix over
K
, and suppose that the characteristic
polynomial of
A
factors into linear factors in
KOExŁ
. Let
T
be the linear
transformation of
K
n
determined by left multiplication by
A
. Thus,
A
is
the matrix of
T
with respect to the standard basis of
K
n
. A second matrix
A
0
is similar to
A
if, and only if,
A
0
is the matrix of
T
with respect to some
other ordered basis of
K
n
. By Theorem
8.7.4
A
is similar to a matrix
A
0
in Jordan canonical form, and the matrix
A
0
is unique up to permutation of
Jordan blocks. W have proved:
Theorem 8.7.6.
(Jordan canonical form for matrices.) Let
A
be an
n
–
by–
n
matrix over
K
, and suppose that the characteristic polynomial of
A
factors into linear factors in
KOExŁ
. Then
A
is similar to a matrix in Jordan
canonical form, and this matrix is unique up to permuation of the Jordan
blocks.
Definition 8.7.7.
A matrix is called the
Jordan canonical form of
A
if
it is in Jordan canonical form, and
it is similar
A
.
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Factors, Multiplication, Jordan Canonical Form, Matn .K/

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