328 B. PRELIMINARIES ON MATRIX RINGS
Proof. Define
F =
A I
B 0
. (B.27)
Then G is the Schur complement of F with respect to A. By Fact B.1.9,
|Ag
JK
= f
N
({n}+J),N ({n}+K),
=
A I
.K
B
J.
0
. (B.28)
Note that each of the last l columns of the minor consists of all zeros except for a single 1. If we
expand this minor about the last l columns using Laplaces expansion (Fact B.1.4),then the expansion
consists of a single term, since the submatrix consisting of the last l columns has only nonzero l × l
minor. Thus, if K ={k
1
, ···,k
l
}, then
|Ag
JK
= (1)
v
(1)
l
A
N\K
B
J.
, (B.29)
where
v =
l
i=1
(n + i) +
l
i=1
k
i
. (B.30)
Now the minor on the right side of (B.29) is not quite in the form stated in Fact B.1.10, since
the rows of B occur below those of A, rather than substituting for them (as in Example B.26, for
instance). It is a matter of detail to verify that the parity factor (1)
v+l
accounts for this difference.
2
Fact B.1.12 Suppose A F
n×n
,B F
n×m
, |A| = 0 and let G = A
1
B. Suppose J
S(l, n), K S(l,m). Then |Ag
JK
equals the determinant of the n × n matrix obtained from
A by replacing the columns in J of A by the columns in K of B.
The proof is similar to that of Fact B.1.10 and is left to the reader.
B.2 CANONICAL FORMS
Let R be a principal ideal domain, and let F be the field of fractions associated with R. In this section,
we study the sets R
n×m
and F
n×m
, consisting of n × m matrices whose elements belong to R and
F, respectively. We prove the existence of two canonical forms, namely the Smith form on R
n×m
and
the Smith-McMillan form on F
n×m
.
A matrix A R
n×m
is a le ft associate of B R
n×m
(denoted by A =
L
B) if there is a unimod-
ular matrix U R
n×n
such that A = UB. A is a right associate of B (denoted by A =
R
B) if there
is a unimodular matrix V R
m×m
such that A = BV . A is equivalent to B (denoted by A B)if
B.2. CANONICAL FORMS 329
there are unimodular matrices U R
n×n
,V R
m×m
. such that A = UBV. It is left to the reader
to verify that =
L
, =
R
, are all equivalence relations.
We now commence our study of canonical forms.
Lemma B.2.1 Suppose a
1
, ···,a
n
R, and let d
n
be a GCD of this set of elements. Then there exists
a matrix P
n
R
n×n
whose first row is [a
1
···a
n
] and whose determinant is d
n
.
Proof. The proof is by induction on n.Ifn = 2, by Theorem A.3.3 there exist p
1
,p
2
R such that
p
1
a
1
+ p
2
a
2
= d
2
. Now let
P
2
=
a
1
a
2
p
2
p
1
(B.1)
For larger values of n, let d
n1
be a GCD of a
1
, ···,a
n1
. By the inductive hypothesis, we can
construct a matrix P
n1
R
n1×n1
whose first row is [a
1
···a
n1
] and whose determinant is
d
n1
. By Fact B.1.3,
n1
j=1
a
j
(1)
1+j
m
1j
(P
n1
) = d
n1
. (B.2)
so that
n1
j=1
a
j
d
n1
(1)
1+j
m
1j
(P
n1
) = 1 . (B.3)
For convenience, let z
j
=−a
j
/d
n1
. Next, by Problem A.3.3, d
n
is a GCD of d
n1
and a
n
.By
Theorem A.3.3, there exist x,y R such that xd
n1
+ ya
n
= d
n
. Now define
P
n
=
a
n
P
n1
0
.
.
.
0
yz
1
··· yz
n1
x
. (B.4)
Expanding |P
n
| about the last column and using (B.3) gives |P
n
|=xd
n1
+ ya
n
= d
n
. 2
Theorem B.2.2 (Hermite Form) Every A R
n×n
is a left associate of a matrix B that is lower-
triangular (i.e., b
ij
= 0 for j>i).
330 B. PRELIMINARIES ON MATRIX RINGS
Proof. Let d
n
be a GCD of {a
1n
, ···,a
nn
}, i.e., the elements of the last column of A. By Theo-
rem A.3.5, there exist elements p
1
, ···,p
n
R such that
n
i=1
p
i
a
in
= d
n
. (B.5)
By Problem A.3.4, 1 is a GCD of the set p
1
, ···,p
n
. Hence, by a slight variation of Lemma B.2.1,
there exists a unimodular matrix U R
n×n
such that its last row is [p
1
···p
n
].Now(U A)
nn
= d
n
.
Also, for i = 1, ···,n 1,(UA)
in
belongs to the ideal generated by a
1n
, ···,a
nn
and is thus a
multiple of d
n
.Letz
i
= (U A)
in
/d
n
for i = 1, ···,n 1, and define
U
n
=
10··· 0 z
1
01··· 0 z
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00··· 1 z
n1
00··· 01
. (B.6)
Then U
n
is unimodular since its determinant is 1. Moreover, U
n
UA is of the form
U
n
UA =
0
A
n1
.
.
.
0
a
n1
d
n
. (B.7)
In other words, the last column of A has been reduced to zero above the diagonal by means of left
multiplication by an appropriate unimodular matrix. One can now repeat the procedure with the
n 1 ×n 1 matrix A
n1
, and eventually arrive at a lower triangular matrix. For clarity, the next
step of the algorithm is briefly outlined: Let d
n1
denote a GCD of the elements of the last column
of A
n1
. As above, there exists a unimodular matrix
¯
U
n1
R
n1×n1
such that
¯
U
n1
A
n1
=
0
A
n2
.
.
.
0
a
n2
d
n1
. (B.8)
Now define the n × n unimodular matrix
U
n1
= Block Diag{
¯
U
n1
, 1} . (B.9)

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