I nternat. J. Math.
Math. Sci.
Vol. 3 No.
(1980) 185-188
185
RESEARCH NOTES
CONTROLLABILITY, BEZOUTIAN AND RELATIVE PRIMENESS
B.N. DATTA
Instituto de Matemtica
Estatlstica e CiSncia da Computago
Universidade Estadual de Campinas
SP
Brasil
Campinas
(Received June II,
ABSTRACT.
Let
1979)
g(x) be two polynomials of degree n. Then it is well- known
f(x) and
if
and
only if f(x) and g(x) are relatively prime. We give an alternative proof of this
re-
that the Bezoutian matrix
Bfg
associated with f(x) and g(x) is nonsingular
sult. The proof is based on a result on controllability derived in this note.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 15A30, 15A63.
KEY WORKS AND PHRASES. Bezoan, Controllability, Nuly
i.
INTRODUCTION
Let
bn_I x
n-2
f(x)
n
x
b2x
n-I
anX
b
1
n-2
an_i x
a2x
I
and
g(x)
n
x
n-I
bnX
be two polynomials of degree n.
Then the Bezoutian bilinear form defined by
B(f g)
a
f(y) g(x)
f(x)g(y)
x-y
f(x)
and
n-I
7.
i ,k-0
bik
x y
g(x)
is given by
186
B.N. DATTA
The symmetric matrix
THEOREM i.
Bfg
(bik)
Bfg
is known as the Bezoutian matrix.
f(x) and
is nonsingular iff
g(x) are relatively prime.
The above result is classical and is well-known. Various proofs of this
result
are available in the literature (for references see the survey of Krien and
Naimark
[] and the paper of Honseholder [3).
In this note, we give
Lemma 1
a proof of this result using the idea of controllability.
that follows forms the main tool of our proof. Besides its
t th proof of theorem i, it is important in its
own right and many
application
find
applica-
tions elsewhere.
2.
TWO LEMMAS ON CONTROLLABILITY.
A pair of matrices (A,B), where A is nn and B is nm, is controllable
nnm matrix C(A,B)
LEMMA I.
(B, AB,
n.
be the companion matrix of
0
I
0
0
I..
""
a2
I
X
has rank
the
Let
0
of
A2B,...,An-IB)
if
f(x) and let
an/
X, with x as its last row, be a
n
solution
ATx.
T
Then X is nonsingular iff
(A
PROOF. Let
be
x 1,
x2,..,
x
n)
x
n
is controllable
rows of X
Then the equation
equivalent to
xA-- hx.
x.A
l
x
i-I
/
.x
i
n.
i
2, 3,...,n.
X
ATx
is
CONTROLLABILITY, BEZOUTIAN AND RELATIVE PRIMENESS
187
The last equations can be written in the form:
x
ax
=xAn
n n
n-i
Xn_ 2
Xn_iA-an_iXn
(XnA anXn)A
xA2n
n-lXn
Xn_ 2
Xn_ 3
axA- a
n n
A-a
=x
An-I
2
(xnA -axAn n
2Xn
n
x A
n
an_2xn
Xl
3
a x A
n n
x2
X
x
n
T, ttT)
--bkl bk2x
+
XnA an-2Xn
-an
0
i
XA
of
ATx
a2
-an_l
a
-a
n
-a
n
a
-a
0
0
0
0
I
0
0
0
i
a2Xn
T
(A
xTn)
3
4
is controllable
A be the same as in (I) and let
Let
H
be given by
. .I
H
/
I
X
is nonsingular if and only if
LEMMA 2 [2].
then, (A
a
n I
An-2 an-I xnAn-3
I
whence, X
2
an-lXn -A
ax
n n
n
Thus, we have, for any solution
x
an_iXn
is controllable if and only if the poly’nornials
/
n-1
bknX
(k
1,...,n)
f(x) and
have no coon -ero.
Pk(X)
188
B.N. DATTA
PROOF OF THEOREM I.
It is shown in [I] that
Bf gA
So, by Lemma I
Bf g
ATBfg
is nonsingular if and only if
last row of the Bezoutian matrix
(AT,hnT)
where
is
h
n
the
is controllable.
Bfg,
It is an easy computation to see that
hn (al bl’
a2
b2’"’’an bn)"
Applying Lemma 2 to the pair [A
only if the polynomials
+
(al-b I)
f(x) and
T
hTn
h(x)
are relatively prime. But, h(x)
tively prime if and only if
REMARK. When
Bf g
f(x) and
we see that
n-I
(an-bn)X +
g(x)
Bfg is nonsingular if and
n-2
+ (a2-b2)x
(an-bn)X +
f(x), and f(x) and g(x) are rela-
h(x) are so.
is singular, f(x) and g(x) have a common zero and in this case,
the degree of g.c.d, is equal to the nullity of the controllability matrix
(hn,
T T
A
hn,
T )2
(A
h
T
n
T
(A
)n-lhnT).
REFERENCES
i. B.N. Datta. On the Routh-Hurwitz-Fujiwara and the Schur-Cohn-Fujiwara theorems
for the root-separation problem, Lin. Alg. Appl.,
22 (1978)
235-246.
2. M.L. Hautus. Controllability and observability conditions for linear
autonomous
systems, Nederel. Akad. Wetensch. Proc. Ser.,A-72 (1969) 443-448.
3. A.S. Householder. Bezoutian, elimination and localization, SlAM
Rev..2(1970)73-78.
4. M.G. Krein and M.A. Naimark. The method of symmetric and hermitian forms
in
the
theory of the separation of the roots of algebraic equations (in Russian), GNT I,
Kharkov, 1936.