A SIMPLE METHOD FOR FINDING THE INVERSE MATRIX OF VANDERMONDE MATRIX. E. A. Rawashdeh. 1. Introduction

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1 MATEMATIČKI VESNIK MATEMATIQKI VESNIK Corrected proof Available online research paper originalni nauqni rad A SIMPLE METHOD FOR FINDING THE INVERSE MATRIX OF VANDERMONDE MATRIX E A Rawashdeh Abstract A siple ethod for coputing the inverse of Vanderonde atrices is presented The inverse is obtained by finding the cofactor atrix of Vanderonde atrices Based on this, it is directly possible to evaluate the deterinant and inverse for ore general Vanderonde atrices Introduction The Vanderonde atrices are an essential topic in applied atheatics, natural science and engineering For exaple, they appear in the fields of nuerical analysis, atheatical finance, statistics, geoetry of curves and control theory (cf, eg, [,3 5,8,9] and references therein) Moreover, Vanderonde atrices have gained uch interest in wireless counications due to their frequent appearance in nuerous applications in signal reconstruction, cognitive radio, physical layer security, and MIMO channel odeling (cf, eg, [6, 7,, ] and references therein) In particular, when Rawashdeh et al [] studied the nuerical stability of collocation ethods for Volteera higher order integro-differential equations, they coputed the eigenvalues of a certain atrix The key point for the evaluation of such eigenvalues is to find the inverse of a Vanderonde atrix Recently, Vanderonde atrices and their inverses play an iportant role to deterine logarithic functions of the sub-syste s density atrices [3] In this paper, we present an explicit forula for finding the inverse of Vanderonde atrices Then we copute the deterinant as well as the inverse of ore general Vanderonde atrices that are obtained by deleting one or two rows and coluns of Vanderonde atrices 00 Matheatics Subject Classification: C0, 5A09, 5A5 Keywords and phrases: Vanderonde atrix; inverse of a atrix; deterinant of a atrix

2 The inverse atrix of Vanderonde atrix Main Results It is known that the Vanderonde atrix is defined by c c c c c c V = V (c,, c ) = c c c and its deterinant is given by (c j c k ) Therefore, if the nubers c, c,, c are distinct, then V is invertible Finding the inverse of the Vanderonde atrix has been investigated by any researchers, for exaple Yiu [4] used a technique based on partial fraction decoposition of a certain rational function to express the inverse of V as a product of two atrices, one of the being a lower triangular atrix L Richard [0] wrote the inverse of the Vanderonde atrix as a product of two triangular atrices F Soto and H Moya [3] showed that V = DW L, where D is a diagonal atrix, W is an upper triangular atrix and L is a lower triangular atrix However, in all of these techniques V is not deterined explicitly In this section we present a new, efficient and easy-to-use ethod for coputing V First we introduce the following notations S k :=S k (c,, c ) = c i c i c ik, for k, We also define i <<i k S 0 :=S 0 (c,, c ) =, and S k = 0 for k / {0,, } S k,j := S k (c,, c j, c j+,, c ), for k, and j It is clear that (x c k ) = ( ) k S k x k k= The following lea can be used to copute the cofactor atrix of the Vanderonde atrices as we will see later Lea Let c, c,, c be real nubers and i {0,,, } Then the deterinant of the atrix c c c i c i+ c V i (c, c,, c ) = c c c i is given by det(v i (c, c,, c )) = S i c c c i c i+ c c i+ (c j c k ) c

3 E A Rawashdeh 3 Proof Define the polynoial f(x) := c c c i c i c i+ c c c c i c i c i+ c c c c i c i c i+ c x x x i x i x i+ x Since f(x) is the deterinant of the Vanderonde atrix V (c, c,, c, x), we have f(x) = (c j c k ) (x c j ) = (c j c k ) ( ) k S k x k j= The coefficient of x i in f(x) is ( ) +i+ det(v i (c, c,, c )) which is equal to ( ) i S i (c j c k ) Hence, det(v i (c, c,, c )) = S i (c j c k ) The proof presented here is short and straightforward, unlike the alternative proof presented by Rawashdeh et al [] based on the atheatical induction on the size of atrices Now we are in the position to find a siple forula for coputing the inverse of V Lea Let c, c,, c be distinct real nubers and c c c c c c V = V (c,, c ) = c c c be the Vanderonde atrix Then the inverse of V is the atrix whose eleents are given by (V ) i,j = ( ) i+j S i,j with l = j or k = j, where i, j =,, l<k Proof It is known that V = Adj(V ) det(v ), where Adj(V ) is the transpose of the cofactor atrix of V Fro Lea, the entries of Adj(V ) are given by (Adj(V )) i,j =( ) i+j det(v i (c,, c j, c j+,, c )) =( ) i+j S i,j l<k(l,k j), where V i (c,, c j, c j+,, c ) is the atrix obtained fro the atrix V by eras-

4 4 The inverse atrix of Vanderonde atrix ing the i-th colun and j-th row Thus (V ) i,j = ( ) i+j l<k(l,k j) l<k l<k S i,j S i,j = ( ) i+j with l = j or k = j, where i =,, and j =,, This copletes the proof c c c 3 Exaple 3 If V = c c c 3 c 3 c 3 c 3, then V has the for: 3 c 4 c 4 c 3 4 c c 3 c 4 (c 4 c )(c 3 c )(c c ) c c 3 +c c 4 +c 3 c 4 (c 4 c )(c 3 c )(c c ) c +c 3 +c 4 (c 4 c )(c 3 c )(c c ) (c 4 c )(c 3 c )(c c ) c c 3 c 4 (c 4 c )(c 3 c )(c c ) c c 3 +c c 4 +c 3 c 4 (c 4 c )(c 3 c )(c c ) c c 3 c 4 (c 4 c )(c 3 c )(c c ) (c 4 c )(c 3 c )(c c ) V i,i,,i (c, c,, c ) = c c c 4 (c 4 c 3 )(c 3 c )(c 3 c ) c c +c c 4 +c c 4 (c 4 c 3 )(c 3 c )(c 3 c ) c +c +c 4 (c 4 c 3 )(c 3 c )(c 3 c ) (c 4 c 3 )(c 3 c )(c 3 c ) c c c 3 (c 4 c 3 )(c 4 c )(c 4 c ) c c +c c 3 +c c 3 (c 4 c 3 )(c 4 c )(c 4 c ) c c c 3 (c 4 c 3 )(c 4 c )(c 4 c ) (c 4 c 3 )(c 4 c )(c 4 c ) In the next lea, we find the deterinant as well as the inverse of ore general Vanderonde atrices of the for c i c i c i c i c i c i c i c i c i where {i, i,, i } is an increasing sequence of non negative integers and satisfying {i, i,, i } {0,,, + } or {i, i,, i } {0,,, + } Lea 4 Let c, c,, c be real nubers (I) The deterinant of the atrix c c c i c i+ c j c j+ c + c c c i c i+ c j c j+ c + V i,j(c, c,, c ) = c c c i c i+ c j c j+ c + is given by det(v i,j (c, c,, c )) = (c j c k )(S i S j+ S i+ S j ) where {i, j} {0,,, + } and i < j (II) The deterinant of the atrix V i,j,r(c, c,, c ) = c c c i c i+ c j c j+ c r c r+ c + c c c i c i+ c j c j+ c r c r+ c + c c c i c i+ c j c j+ c r c r+ c +,

5 E A Rawashdeh 5 is given by det(v i,j,r(c, c,, c )) = (c j c k )((S is j+ S i+s j)s r+) (S is j+ S i+s j)s r+ + (S i+s j+ S i+s j+)s r) where {i, j, r} {0,,, + } and i < j < r Proof (I) Define the polynoial c c c i c i+ c j c j c j+ c + c c c i c i+ c j c j c j+ c + f(x) := c c c i c i+ c j c j c j+ c + x x x i x i+ x j x j x j+ x + Then it is clear that f(x) = (a x + a 0 ) ( ) k S k x k The leading coefficient of f(x) is a and fro Lea, we have a = det(v i (c, c,, c )) = S i (c j c k ) The constant ter of f(x) is ( ) S a 0 which is equal to ( ) + det(b), where c c c i c i+ c j c j c j+ c + c c c i c i+ c j c j c j+ c + B = c c c i c i+ c j c j c j+ c + Again fro Lea, we have det(b) = c c c det(v i (c, c,, c )) = S S i+ (c j c k ), thus a 0 = S i+ (c j c k ) Hence, we have f(x) = (c j c k ) ( ) k S k x k (S i x + S i+ ) Since ( ) ++j det(v i,j (c, c,, c )) is the coefficient of x j in f(x) which is (c j c k )(( ) j S i+ S j + ( ) j+ S i S j+ ), we obtain det(v i,j (c, c,, c )) = (c j c k )(S i S j+ S i+ S j ) (II) Define the polynoial f(x) = det(v i,j (c, c,, c, x)) Then f(x) is a polyno-

6 6 The inverse atrix of Vanderonde atrix ial of degree less than or equal to + and it is clear that f(x) = (a x + a x + a 0 ) ( ) k S k x k The leading coefficient of f(x) is a and fro part (I), we have a = det(v i,j (c, c,, c )) = (S i S j+ S i+ S j ) The constant ter of f(x) is ( ) S a 0 which is equal to ( ) + S det(v i,j (c, c,, c )), so fro part (I), a 0 = (c j c k )(S i+ S j+ S i+ S j+ ) Now the coefficient of x j is zero, so (c j c k ) ( ) j+ a S j+ + ( ) j+ a S j+ + ( ) j a 0 S j = 0, and substituting the values of a and a 0, yields (c j c k ) ( (S i S j+ S i+ S j )S j+ ) + (S i+ S j+ S i+ S j+ )S j a S j+ = 0 Thus we have a = (c j c k )(S i S j+ S i+ S j ) Hence, f(x) = (c j c k ) ( ) k S k x k( (S i S j+ S i+ S j )x ) +(S i S j+ S i+ S j )x + (S i+ S j+ S i+ S j+ ) Since ( ) ++r det(v i,j,r (c, c,, c ) is the coefficient of x r in f(x) which is we obtain ( ) r+ a S r+ + ( ) r+ a S r+ + ( ) r a 0 S r, det(v i,j,r (c, c,, c )) = ( (c j c k ) (S i S j+ S i+ S j )S r+ (S i S j+ S i+ S j )S r+ +(S i+ S j+ S i+ S j+ )S r ) Reark 5 Leas and 4 can be used to find the deterinants and cofactor atrices of the atrices V i (c,, c ) and V i,j (c,, c ) Thus coputing the inverses of these atrices is easy to deterine, provided that V i (c,, c ) and V i,j (c,, c ) are invertible However, finding the cofactor atrix as well as the inverse of the atrix V i,j,r (c,, c ) are still needed to be evaluated In view of

7 E A Rawashdeh 7 Leas and 4, it will be interesting to find the inverse of the atrix c i c i c i V i,i,,i (c, c,, c ) = c i c i c i c i c i c i where {i, i,, i } is an increasing sequence of non-negative integers, which we plan to discuss in a forthcoing paper References [] JM Corcuera, D Nualart, W Schoutens, Copletion of a Lévy arket by power-jup assets, Finance Stochastics, 9() (005), 09 7 [] E Rawashdeh, D Mcdowell, L, Rakesh, The Stability of Collocation Methods for Higher-Order Volterra Integro-Differential Equations, IJMMS, 005: (005) [3] A Eisinberg, G Fedele, On the inversion of the Vanderonde atrix, Appl Math Coput, 74 (006), [4] M Goldwur, V Lonati, Pattern statistics and Vanderonde atrices, Theor Coput Sci, 356 (006), [5] H Hakopian, K Jetter, G Zierann, Vanderonde atrices for intersection points of curves, J Approx, (009), 67 8 [6] B Khan, M Debbah, T Y Al-Naffouri, O Ryan, Estiation of the distribution deployent of sensor networks, in Proceedings of the International Syposiu on Inforation Theory, ISIT009, (009) [7] M Kobayashi, M Debbah, On the secrecy capacity of frequency-selective fading channels: A practical Vanderonde precoding, in PIMRC, Cannes, France, Septeber (008) [8] R L de Lacerda Neto, L Sapaio, H Hoffsteter, M Debbah, D Gesbert, R Knopp, Capacity of MIMO systes: Ipact of polarization, obility and environent, in IRAMUS Workshop, Val Thorens, France, January (007) [9] R Norberg, On the Vanderonde Matrix and its Role in Matheatical Finance, Laboratory of Actuarial Matheatics (working paper), University of Copenhagen, (999) [0] L Richard Turner, Inverse of the Vanderonde atrix with applications, NASA TN D-3547 [] O Ryan, M Debbah, Convolution operations arising fro Vanderonde atrices, IEEE Transactions on Inforation Theory, 57(7) (0), [] L Sapaio, M Kobayashi, O Ryan, M Debbah, Vanderonde frequency division ultiplexing, 9th IEEE Workshop on Signal Processing Advances for wireless applications, Recife, Brazil, (008) [3] F Soto-Eguibar, H Moya-Cessa, Inverse of the Vanderonde and Vanderonde confluent atrices, Appl Math Inforation Sci, 5(3) (0), [4] Yiu-Kwong Man, On the Inversion of Vanderonde Matrices, Proceedings of the World Congress on Engineering 04 Vol II, WCE 04, July - 4, 04, London, UK, (received 30907; in revised for 00308; available online 40708) Departent of Matheatics, Yarouk University, Irbid-Jordan E-ail: edris@yuedujo