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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