Download Greatest Common Divisor of Two Polynomials Let a@) = A” + ay +

LINEAR
ALGEBRA
Greatest
AND
Common
ITS
APPLICATIONS
Divisor
of Two Polynomials
S. BARNETT
University
of
Loughborough,
Communicated
Technology
Leicestershive,
England
by L. Mirsky
ABSTRACT
If a(n) and b(l)
are two polynomials and A is the companion matrix of a(n),
K, where k is the degree of the greatest common
then the matrix b(A) has rank 6(a) divisor of a(1) and b(i).
It is shown that, if the first k columns of b(A) are expressed
as linear combinations of the remaining 6(a) - k columns (which are linearly independent), then the greatest common divisor is given by the coefficients of column
k +
1 in these expressions.
Let
a@) = A” +
ay
+ . *. + an
and
b(A) = Am + b,;l”-’
be two polynomials
+ . . . $ 6,
over some field and let
r
0
1
0
0
0
1
0
0
0
A=
-
a,
-
a,_,
Linear
Copyright
0
-
a n-2
Algebra
1970 by American
and
Its
Elsevier
Applications
Publishing
3(1970),
Company,
7-9
Inc.
8
S. BARNETT
be the companion
results
matrix
on polynomial
immediately
of a(A).
matrices
Then
by specializing
given previously
more
general
[l, 21 the following
is
deduced :
1. a(l) and b(l) are relatively prime if and only if the matrix
THEOREM
R = b(A) z A”
+ b,A”-’
+ -. - + b,I,
is nonsingular.
Furthermore,
the degree of the greatest common divisor (g.c.d.) of a(n) and b(i) is equal
to n-rank
R.
In fact
ni
jli)%‘, then
(1 -
once.
that
it is not
difficult
to prove
b(A) = n
(A -
The second part is derived
rank
b(A) = rank n
Theorem
If b(l) =
so the first part
&I,)‘;
by putting
;liIn)a”, and
(J -
1 directly:
follows
at
A into Jordan
form J, SO
using
that
the
fact
A is
nonderogatory.
The purpose
be determined
of the present
note is to show how the g.c.d. may itself
from the matrix
Let
R.
a(A)
= P+ ap
be the g.c.d.
of a(l)
THEOREM
and if
yi
C;=k+l
xijrj,
i
of R, then dp = x,+,_~,,+~,
First,
Proof.
that
the first
k columns of R are linearly
independent,
1, 2, . . . , k, where ri devtotes the ith column
=
p = 1, 2,. . ., k.
let b(L) = c(l)d(il) and consider d(A) = Ak + d,Ak-l
By Theorem
* . - + d,I,.
Then:
The last n -
2.
=
and b(l).
+ ...+ a,
n -
1, d(A) has rank n -
k rows of d(A)
+
k, and it is easily verified
are
aA akpl ... a, a, i 0 0 ... 0 0
0
i
Clearly
a,
a, a, a, i
.
.
.
.
.
.
.
.
.
0
_
...
the last
n -
.
.
.
.
.
.
.
k columns
si =
i
.
.
.
of d(A)
yijsj,
0
.
.
.
.
0
.
.
.
.
.
.
.
.
.
...
are linearly
i = 1,2,. . . , k,
j=k+l
Linear
Algebra
and
Its
Applications
8(1970),
7-9
0
.
. 1.
(1)
.
a, i 1
independent.
If
(2)
GREATEST
COM;LlON
where si stands
DIVISOR
OF
for the ith column
row of (1) gives ~~,~+r = dk_i+I,
Next,
Theorem
since
of d(A), then inspection
whence
particular
3++r
= dh_-lfl,
The result of Theorem
prime,
it follows
Now ri = c(A)s,,
independent,
ri = ~~=li+,
of the first
i = 1, 2,. . . , R.
c(A) is nonsingular.
so rk_kr, . . . I Y, are linearly
yijc(A)sj,
9
POLYNOMIALS
a(A) and c(A) are relatively
1) that
algorithm
TWO
and from (2) c(A)si =
Thus
yijyj.
xij = yij,
(again
by
i = 1, 2, . . . , n
cyzk+,
all i, j,
.
and in
i = 1, 2,. . ., k.
2 provides an interesting
or the determinantal
expression
alternative
to Euclid’s
for the g.c.d. of two polynomials.
REFERENCES
1 S. Barn&t,
Pvoc.
Regular
Cambridge
2 S. Barn&t,
regular
Received
Degrees
polynomial
October
polynomial
Philos.
of
Sm.
greatest
matrices.
14.
matrices
65(1969),
having
common
Proc.
relatively
prime
determinants.
585-590.
divisors
Cambridge
of
Philos.
invariant
Sot.
factors
65(1969),
to
of
two
appear.
1968
Linear
Algebra
and
Its
Applications
3(19’70),
7-9