A property of the elementary symmetric functions

Transcription

1 Calcolo manuscrpt No. (wll be nserted by the edtor) A property of the elementary symmetrc functons A. Esnberg, G. Fedele Dp. Elettronca Informatca e Sstemstca, Unverstà degl Stud della Calabra, 87036, Rende (Cs), Italy, e-mal: esnberg@uncal.t, fedele@s.des.uncal.t Receved: date / Revsed verson: date Abstract. We show a smple relaton concernng the elementary symmetrc functon and present two applcatons concernng the nverse of a Vandermonde matrx and the spectral propertes of square matrces. 1. Introducton Let n be a postve nteger and consder the set of complex numbers X n = {x 1, x 2,..., x n }. Defne the polynomal S n (x) = n =1 (x x ) of degree n, such that S n (x) = n =0 ( 1)n+ σ(n, n )x, where σ(n, k) = x π1 x π2...x πk (1) 1 π 1 <...<π k n s the kth order elementary symmetrc functon assocated wth X n. In ths short note we show an dentty concernng the functons σ(n, k) and φ(n, k) = ( 1) n+k S n(x k ), then we apply t for obtanng n a smple way the nverse structure of a nonsngular Vandermonde matrx V n. Our formulaton, whch we present n secton 3, can be derved from [8], but n our opnon t s more flexble and allows one to obtan specal algorthms for partcular sets of nodes [2], [3], [5]. Moreover, relyng on ths dentty n [4] we desgn an accurate and fast algorthm for the nverson of V n that provdes better performance than the algorthm of [7]. Another applcaton of ths dentty s the smple proof of a result of [6] where the egenvectors of a matrx are expressed n terms of ts

2 2 A. Esnberg, G. Fedele egenvalues and of the elementary symmetrc functons. We present ths result n secton Man result By usng an nducton argument t s a smple matter to prove that σ(n, s) = σ(n 1, s) + x n σ(n 1, s 1), 0 < s n, σ(n, s) = 0, s > n, σ(n, 0) = 1, n 0, and that φ(n + 1, s) = φ(n, s)(x n+1 x s ), 1 s n, φ(n + 1, n + 1) = n (x n+1 x k ), n 1, k=1 φ(2, 1) = φ(2, 2) = x 2 x 1. Consder the functon (2) (3) h(n,, j, q) = ( 1) k+r x k+q x r jσ(n, r)+( 1) j+1 x q+1 j φ(n, j)δ,j. (4) We have the followng result Theorem 1. For n 2 and q 1 one has h(n,, j, q) = 0,, j = 1, 2,..., n. Proof. We proceed by nducton on n. For n = 2 we have h(2,, j, q) = 0 by drect nspecton. For the nductve step, performng standard algebrac manpulatons we obtan the followng equvalent expressons for h(n + 1,, j, q): h(n+1,, j, q) = (x n+1 x )h(n,, j, q)+( 1) j+1 (x x j)x q+1 j φ(n, j)δ,j,, j = 1, 2,..., n, (5) h(n + 1,, j, q) = (x n+1 x j)h(n,, j, q),, j = 1, 2,..., n. (6) Hence we obtan that, f h(n,, j, q) = 0 for, j = 1, 2,..., n then h(n + 1,, j, q) = 0 for, j = 1, 2,..., n. Consder the case where =

3 A property of the elementary symmetrc functons 3 n + 1 or j = n + 1 and use the followng equvalent expresson for h(n + 1,, j, q): h(n + 1,, j, q) = (x n+1 x n ) ( 1) r+k x k+q x r jσ(n, r) ( 1) r x r jσ(n, n r) + ( 1) j+1 x q+1 φ(n + 1, j)δ,j, x q+1, j = 1, 2,..., n + 1, j (7) h(n + 1,, j, q) = (x n+1 x n j) ( 1) r+k x k+q x r jσ(n, r) ( 1) r x r σ(n, n r) + ( 1) j+1 x q+1 φ(n + 1, j)δ,j, Snce x q+1, j = 1, 2,..., n + 1. j (8) ( 1) r x r tσ(n, n r) = ( 1) n φ(n + 1, t)δ n+1,t, t = 1, 2,..., n + 1, (9) then, usng (7) and (8) we have that h(n + 1, n + 1, j, q) = 0 for j = 1, 2,..., n + 1 and h(n + 1,, n + 1, q) = 0 for = 1, 2,..., n + 1 respectvely. 3. Applcatons 3.1. Egensystems The followng theorem gves an explct expresson for a system of a matrx egenvectors when the egenvalues are known. Theorem 2. Let A be a n n matrx wth dstnct egenvalues Λ = {α 1, α 2,..., α n } where α k C, k = 1, 2,..., n and U = {u 1, u 2,..., u n }, wth u k C n, k = 1, 2,..., n, the correspondng rght-egenvectors. A system of rght-egenvectors s w = ( 1) r+k+1 σ(n, n k r)α k 1 A r h, = 1, 2,..., n (10) where h C n ; h 0; h T u k 0, k = 1, 2,..., n.

4 4 A. Esnberg, G. Fedele Proof. Let Note that h = γ j u j. (11) j=1 ( h T u k 0, k = 1, 2,..., n ) (γ k 0, k = 1, 2,..., n) (12) and φ(n, k) 0, k = 1, 2,..., n (13) due to the fact that α α j, j;, j = 1, 2,..., n. Therefore w j can be expressed as: where w j = γ j θ(n,, j)u j, j = 1, 2,..., n (14) j=1 θ(n,, j) = ( 1) r+k+1 α k 1 αjσ(n, r n k r),, j = 1, 2,..., n. By substtutng q = 1 n theorem 1, we have: (15) θ(n,, j) = ( 1) j+1 φ(n, j)δ,j,, j = 1, 2,..., n. (16) Therefore we have: w = ( 1) +1 γ φ(n, )u, = 1, 2,..., n, (17) and (10) follows The Vandermonde matrx Let X n = {x 1, x 2,..., x n } be a set of n dstnct complex numbers, usually called nodes. Vandermonde matrx V n s defned as the n n matrx whose generc element v n (, j) s gven by: v n (, j) = x j 1,, j = 1, 2,..., n. (18) The followng theorem gves the explct expresson for the generc elements of the nverse of V n.

5 A property of the elementary symmetrc functons 5 Theorem 3. Let W n be the nverse of V n. Then the generc element w n (, j) of W n s: w n (, j) = ψ(n,, j),, j = 1, 2,..., n (19) φ(n, j) where n ψ(n,, j) = ( 1) +j ( 1) r x r jσ(n, n r),, j = 1, 2,..., n. (20) Proof. It must be shown that But (V n W n ),j = δ,j,, j = 1, 2,..., n. (21) (V n W n ),j = ( 1) k+r+j x k 1 x r 1 jσ(n, n k r) φ(n, j) (22) therefore, by substtutng q = 1 n theorem 1 the (21) follows. Note that the (19) gves also the followng factorzaton of W n : where W n = M V T n D (23) M(, j) = ( 1) +j+1 σ(n, n+1 j), = 1, 2,..., n; j = 1, 2,...n+1, (24) { } ( 1) D = dag. (25) φ(n, ) =1,2,...,n Acknowledgements We wsh to thank Prof. D. Bn of Unverstá d Psa, Italy, for the helpful dscussons. We also lke to acknowledge the constructve crtcsm of the referee.

6 6 A. Esnberg, G. Fedele References 1. Esnberg, A., Pcard, C., On the nverson of Vandermonde matrx, Proc. of the 8th Trennal IFAC World Congress, Kyoto, Japan, Esnberg, A., Franzé, G., Puglese, P., Vandermonde matrces on nteger nodes, Numer. Math. 80 (1998) Esnberg, A., Franzé, G., Puglese, P., Vandermonde matrces on Chebyshev ponts, Lnear Algrebra and Appl. 283 (1998) Esnberg, A., Fedele, G., On the nverson of the Vandermonde matrx, Not publshed. 5. Esnberg, A, Fedele, G., The Vandermonde matrx on Gauss-Lobatto Chebyshev nodes, Not publshed. 6. Faddeev, D. K., Faddeeva, V. N., Computatonal Methods of Lnear Algebra. Freeman, Gohberg, I., Olshevsky, V., The fast generalzed Parker-Traub algorthm for nverson of Vandermonde and related matrces, Journal of Complexty, 13 (1997), Gohberg, I., Olshevsky, V., Fast nverson of Chebyshev-Vandermonde matrces, Numer. Math. 67 (1994) Traub, J., Assocated polynomals and unform methods for the soluton of lnear problems, SIAM Revew 8, 3 (1966)