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Some Properties of the Q-Adic Vandermonde Matrix

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Some Properties of the Q-Adic Vandermonde Matrix The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 1, pp. 18-33, June 1996. ELA ISSN 1081-3810. SOME PROPERTIES OF THE Q-ADIC VANDERMONDE MATRIX y y VAIDYANATH MANI AND ROBERTE.HARTWIG Abstract. The Vandermonde and con uentVandermonde matrices are of fundamental signi cance in matrix theory. A further generalization of the Vandermonde matrix called the q -adic co ecient matrix was intro duced in [V. Mani and R. E. Hartwig, Lin. Algebra Appl., to app ear]. It was demonstrated there that the q -adic co ecient matrix reduces the Bezout matrix of two p olynomial s by congruence. This extended the work of Chen, Fuhrman, and Sansigre among others. In this pap er, some imp ortant prop erties of the q -adic co ecient matrix are studied. It is shown that the determinant of this matrix is a pro duct of resultants like the Vandermonde matrix. The Wronskian-like blo ck structure of the q -adic co ecient matrix is also explored using a mo di ed de nition of the partial derivative op erator. Key words. Vandermonde, Hermite Interp olation, q -adic expansion. AMSMOS sub ject classi cation. Primary 15A15, 15A21, 41A10, 12Y05 1. Intro duction. Undoubtedly, the Vandermonde matrix is one of the most imp ortant matrices in applied matrix theory. Its generalization to the con uentVandermonde matrix has numerous applications in engineering. This concept was further generalized to the q -adic Vandermonde matrix in [9], ex- tending the work of Chen [4], Fuhrman [7] and Sansigre [3] among others. In this pap er, weinvestigate some of the prop erties of the q -adic Vandermonde matrix. In particular, we shall evaluate its determinant and analyze its blo ck structure. The former extends the Vandermonde determinant, while the lat- ter shows a remarkable analogy to the layered Wronskian structure of the con uentVandermonde matrix. It is well known that the con uentVandermonde matrix for the p olynomial s Y m i x= x ; i i=1 P deg = m = n, can b e constructed by forming the Taylor expansion of i Received by the editors on 5 Decemb er 1995. Final manuscript accepted on 29 May 1996. Handling editor: Daniel Hershkowitz. y North Carolina State University, Raleigh, North Carolina 27695-8205, USA [email protected], [email protected]. 18 ELA Prop erties of q -adic Vandermonde Matrix 19 t x in terms of the p owers of x , i.e., i 1 = 1 x = +1x i i 2 2 2 x = +2 x +1x i i i i . 1 t1 t t x + ::: x = + t i i i . P n 1 nj n n1 j x = x : i j =1 i j 1 If we set x = q and denote the column containing the rst m co ecients i i i t t in the ab ove expansion of x by[x ] , then q ;m i i 3 2 t i t1 7 6 t i 7 6 7 6 t . 2 : [x ] = 7 6 q ;m i i . 7 6 5 4 t tm +1 i i m 1 i m 1 i In case t<m 1, we of course replace the elements involving negativepowers i of by zero. Collecting these co ordinate columns, wemay construct the nm i i n matrix as i m i i h n T n1 [1] [x] ::: [x ] : = i q ;m q ;m q ;m m i i i i i i i n In other words, a is made up of rst m columns of Caratheo dory's n n i m i n1;n1 k k j . Lastly, the n n con uent binomial matrix a= a n j k;j =0 Vandermonde is de ned by matrix n n n 3 : ; ::; ; = s 2 1 m m m s 2 1 To extend this concept further, we recall the q -adic expansion of a given p olynomial f x relative to a monic p olynomial q x. Indeed, the following lemma from [2] is elementary. Lemma 1.1. Let q x be a monic polynomial of degree l over F.Iff x is any polynomial over F, then there exist unique polynomials r x over F,of j degree less than l , such that k X j f x= r x[q x] j j =0 ELA 20 Vaidyanath Mani and Rob ert E. Hartwig for some nite non-negative integer k . Let us now use the ab ove lemma to de ne the q -adic co ordinate column of k a p olynomial. For an arbitrary p olynomial f x= a +a x+:::+a x , its stan- 0 1 k n1 dard co ordinate column relative to the standard basis B = f1;x;:::;x g ST is de ned and denoted by 3 2 a 0 7 6 a 1 7 6 : f =[f ] = 7 6 . st . 5 4 . a k Now for a given pair q; m made up of a p olynomial q x of degree l and a p ositiveinteger m, the q-adic expansion of f xmay b e written as m1 m f x= r x+r xq + :::+ r xq + q ?; 0 1 m1 m where ? denotes the p olynomial quotient after division by q . The q-adic co ordinate column asso ciated with f x for the pair q; misnow de ned as 2 3 r 0 6 7 r 1 6 7 [f ] = ; 6 7 . q;m . 4 5 . r m1 lm1 where r is the standard co ordinate column for r x. It should b e noted that i i m i if the degree of f x is larger than q x ,we simply ignore the quotient m after division by q . m ii the statements q jf and [f ] = 0 are equivalent, i.e., q;m m 4 [f ] =0 , q jf: q;m iii if deg f = k , then [f ] is a k +1 1 column vector. st th Throughout this pap er, we shall denote the i unit column vector by e . i 1.1. The q -adic Vandermonde matrix. We are now ready for the construction of the q -adic Vandermonde matrix. In order to rep eat the con- struction given in 1 over elds which are not algebraically closed, let x b e a foundation p olynomial with the factorization s Y m i x= q i i=1 P s with deg q =l 1 and deg = m l = n. Again, using the q-adic i i i i i=1 t expansion of the standard basis elements x ,wehave m 1 i X t t j m i x = r xq x + q x ?: i i ij j =0 ELA Prop erties of q -adic Vandermonde Matrix 21 t The q -adic co ordinate column [x ] maynow b e formed as i q ;m i i 2 3 t r i0 6 7 t 6 7 r 6 7 i1 t [x ] = 5 ; 6 7 . q ;m i i . 6 7 . 4 5 t r i;m 1 i t t where r is the standard co ordinate column for r x. Lastly, collecting ij ij co ordinate columns, we set i h n1 [1] [x] ::: [x ] 6 W = i q ;m q ;m q ;m i i i i i i l m n i i and construct the n nq-adic Vandermonde matrix W as 2 3 W 1 6 7 W 2 T 6 7 W = 4 5 ::: W s It is clear that when the q 's are linear p olynomials then, 2 reduces to 6 and i T T W to .We shall as such refer to the matrix W as the q -adic Vandermonde matrix. If W is nonsingular, it solves the q -adic interp olation problem for p olynomials as discussed in [9]. Moreover, for any p olynomial f x with degree less than n and standard co ordinate column [f ] , it is clear from 5 that st 7 W [f ] =[f ] ; i st q ;m i i In other words, W acts as a change of basis matrix from the standard to the i q -adic co ordinates. This is a crucial observation which will b e used in the i next section. To analyze W further, we need the relation b etween the q-adic co ordinate k columns of [f ] and [ f ] . This relation is b est handled via the concept q;m q;m of a p olynomial in a hyp ercompanion matrix whichwe shall now address. 1.2. Some basic results hyp ercompanion matrices. If q x= a + 0 l a x + ::: + x is a monic p olynomial of degree l , then the hyp ercompanion 1 m matrix for q x is the m m blo ck matrix 2 3 L 0 : :: 0 q 6 7 N L 0 :: 0 q 6 7 6 7 m : : : : 0 H = 8 ; q 6 7 4 5 0 0 N L 0 q 0 0 0 N L q ELA 22 Vaidyanath Mani and Rob ert E. Hartwig T and L is the companion matrix of q x de ned by where N = E = e e q 1;l 1 l 3 2 0 0 0 ::: a 0 7 6 1 0 0 ::: a 1 7 6 7 6 0 1 0 ::: :: 7 6 : L = q 7 6 : : : ::: :: 7 6 5 4 : : : ::: a k 2 0 0 0 ::1 a k 1 The key result whichwe need is the following lemma dealing with p olynomials of hyp ercompanion matrices.
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