372
8. MODULES
k
D
min
f
m; n
g
, write
A
D
diag
.d
1
; d
2
; : : : ; d
k
/
if
A
is diagonal and
a
i;i
D
d
i
for
1
i
k
.
Proposition 8.4.6.
Let
A
be an
m
–by–
n
matrix over
R
. Then there exist
invertible matrices
P
2
Mat
m
.R/
and
Q
2
Mat
n
.R/
such that
PAQ
D
diag
.d
1
; d
2
; : : : ; d
s
; 0; : : : ; 0/
, where
d
i
divides
d
j
for
i
j
.
The matrix
PAQ
D
diag
.d
1
; d
2
; : : : ; d
s
; 0; : : : ; 0/
, where
d
i
divides
d
j
for
i
j
is called the
Smith normal form
of
A
.
1
Diagonalization of the matrix
A
is accomplished by a version of Gauss
ian elimination (row and column reduction). For the sake of completeness,
we will discuss the diagonalization process for matrices over an arbitrary
principal ideal domain. However, we also want to pay particular attention
to the case that
R
is a Euclidean domain, for two reasons. First, in appli
cations we will be interested exclusively in the case that
R
is Euclidean.
Second, if
R
is Euclidean, Gaussian elimination is a constructive process,
assuming that Euclidean division with remainder is constructive. (For a
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Matrices, Ring, Invertible matrix, Elementary Row Operation, Principal ideal domain

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