MATEMATICKIˇ VESNIK MATEMATIQKI VESNIK research paper 71, 3 (2019), 207–213 originalni nauqni rad September 2019
A SIMPLE METHOD FOR FINDING THE INVERSE MATRIX OF VANDERMONDE MATRIX
E. A. Rawashdeh
Abstract. A simple method for computing the inverse of Vandermonde matrices is presented. The inverse is obtained by finding the cofactor matrix of Vandermonde matrices. Based on this, it is directly possible to evaluate the determinant and inverse for more general Vandermonde matrices.
1. Introduction
The Vandermonde matrices are an essential topic in applied mathematics, natu- ral science and engineering. For example, they appear in the fields of numerical analysis, mathematical finance, statistics, geometry of curves and control theory (cf, e.g., [1,3–5,8,9] and references therein). Moreover, Vandermonde matrices have gained much interest in wireless communications due to their frequent appearance in numer- ous applications in signal reconstruction, cognitive radio, physical layer security, and MIMO channel modeling (cf, e.g., [6,7,11, 12] and references therein). In particular, when Rawashdeh et al. [2] studied the numerical stability of colloca- tion methods for Volteera higher order integro-differential equations, they computed the eigenvalues of a certain matrix. The key point for the evaluation of such eigenval- ues is to find the inverse of a Vandermonde matrix. Recently, Vandermonde matrices and their inverses play an important role to determine logarithmic functions of the sub-system’s density matrices [13]. In this paper, we present an explicit formula for finding the inverse of Vandermonde matrices. Then we compute the determinant as well as the inverse of more general Vandermonde matrices that are obtained by deleting one or two rows and columns of Vandermonde matrices.
2010 Mathematics Subject Classification: 11C20, 15A09, 15A15 Keywords and phrases: Vandermonde matrix; inverse of a matrix; determinant of a matrix.
207 208 The inverse matrix of Vandermonde matrix
2. Main Results
It is known that the Vandermonde matrix is defined by 2 m−1 1 c1 c1 . . . c1 2 m−1 1 c2 c2 . . . c2 V = V (c , . . . , c ) = 1 m . . . . ...... 2 m−1 1 cm cm . . . cm m Q and its determinant is given by (cj − ck). 1≤k
1≤i1<... S0 :=S0(c1, ..., cm) = 1, and Sk = 0 for k∈ / {0, . . . , m}. We also define Sk,j := Sk(c1, ..., cj−1, cj+1, ..., cm), for 1 ≤ k ≤ m − 1, and 1 ≤ j ≤ m. m m Q P m−k k It is clear that (x − ck) = (−1) Sm−kx . k=1 k=0 The following lemma can be used to compute the cofactor matrix of the Vander- monde matrices as we will see later. Lemma 2.1. Let c1, c2, . . . , cm be real numbers and i ∈ {0, 1, . . . , m}. Then the deter- minant of the matrix 2 i−1 i+1 m 1 c1 c1 . . . c1 c1 . . . c1 2 i−1 i+1 m 1 c2 c2 . . . c2 c2 . . . c2 V (c , c , . . . , c ) = i 1 2 m ...... ...... 2 i−1 i+1 m 1 cm cm . . . cm cm . . . cm m Q is given by det(Vi(c1, c2, . . . , cm)) = Sm−i (cj − ck). 1≤k Proof. Define the polynomial 2 i−1 i i+1 m 1 c1 c1 . . . c1 c1 c1 . . . c1 2 i−1 i i+1 m 1 c2 c . . . c c c . . . c 2 2 2 2 2 ...... f(x) := ...... 1 c c2 . . . ci−1 ci ci+1 . . . cm m m m m m m 1 x x2 . . . xi−1 xi xi+1 . . . xm Since f(x) is the determinant of the Vandermonde matrix V (c1, c2, . . . , cm, x), we have m m m m Y Y Y X m−k k f(x) = (cj − ck) (x − cj) = (cj − ck) (−1) Sm−kx . k The proof presented here is short and straightforward, unlike the alternative proof presented by Rawashdeh et al. [2] based on the mathematical induction on the size of matrices. Now we are in the position to find a simple formula for computing the inverse of V . Lemma 2.2. Let c1, c2, . . . , cm be distinct real numbers and 2 m−1 1 c1 c1 . . . c1 2 m−1 1 c2 c2 . . . c2 V = V (c , . . . , c ) = 1 m . . . . ...... 2 m 1 cm cm . . . cm−1 be the Vandermonde matrix. Then the inverse of V is the matrix whose elements are given by −1 i+j Sm−i,j (V )i,j = (−1) m with l = j or k = j, where i, j = 1, . . . , m. Q (ck − cl) l −1 Adj(V ) Proof. It is known that V = det(V ) , where Adj(V ) is the transpose of the cofactor matrix of V . From Lemma 2.1, the entries of Adj(V ) are given by i+j (Adj(V ))i,j =(−1) det(Vi(c1, . . . , cj−1, cj+1, . . . , cm)) m i+j Y =(−1) Sm−i,j (ck − cl), l c2c3c4 −c1c3c4 c1c2c4 −c1c2c3 (c4−c1)(c3−c1)(c2−c1) (c4−c2)(c3−c2)(c2−c1) (c4−c3)(c3−c2)(c3−c1) (c4−c3)(c4−c2)(c4−c1) −c2c3+c2c4+c3c4 c1c3+c1c4+c3c4 −c1c2+c1c4+c2c4 c1c2+c1c3+c2c3 (c4−c1)(c3−c1)(c2−c1) (c4−c2)(c3−c2)(c2−c1) (c4−c3)(c3−c2)(c3−c1) (c4−c3)(c4−c2)(c4−c1) . c2+c3+c4 −c1−c3−c4 c1+c2+c4 −c1−c2−c3 (c4−c1)(c3−c1)(c2−c1) (c4−c2)(c3−c2)(c2−c1) (c4−c3)(c3−c2)(c3−c1) (c4−c3)(c4−c2)(c4−c1) −1 1 −1 1 (c4−c1)(c3−c1)(c2−c1) (c4−c2)(c3−c2)(c2−c1) (c4−c3)(c3−c2)(c3−c1) (c4−c3)(c4−c2)(c4−c1) In the next lemma, we find the determinant as well as the inverse of more general Vandermonde matrices of the form i1 i2 im c1 c1 . . . c1 i1 i2 im c2 c2 . . . c2 V (c , c , . . . , c ) = , i1,i2,...,im 1 2 m . . . ...... i1 i2 im cm cm . . . cm where {i1, i2, . . . , im} is an increasing sequence of non negative integers and satisfying {i1, i2, . . . , im} ⊆ {0, 1, . . . , m + 1} or {i1, i2, . . . , im} ⊆ {0, 1, . . . , m + 2}. Lemma 2.4. Let c1, c2, . . . , cm be real numbers. (I) The determinant of the matrix 2 i−1 i+1 j−1 j+1 m+1 1 c1 c1 . . . c1 c1 . . . c1 c1 . . . c1 2 i−1 i+1 j−1 j+1 m+1 1 c2 c2 . . . c2 c2 . . . c2 c2 . . . c2 Vi,j (c1, c2, . . . , cm) = ...... ...... 2 i−1 i+1 j−1 j+1 m+1 1 cm cm . . . cm cm . . . cm cm . . . cm m Q is given by det(Vi,j(c1, c2, . . . , cm)) = (cj −ck)(Sm−iSm−j+1−Sm−i+1Sm−j) where k Vi,j,r(c1, c2, . . . , cm) = 2 i−1 i+1 j−1 j+1 r−1 r+1 m+2 1 c1 c1 . . . c1 c1 . . . c1 c1 . . . c1 c1 . . . c1 2 i−1 i+1 j−1 j+1 r−1 r+1 m+2 1 c2 c2 . . . c2 c2 . . . c2 c2 . . . c2 c2 . . . c2 ...... ...... 2 i−1 i+1 j−1 j+1 r−1 r+1 m+2 1 cm cm . . . cm cm . . . cm cm . . . cm cm . . . cm E. A. Rawashdeh 211 is given by m Y det(Vi,j,r(c1, c2, . . . , cm)) = (cj − ck)((Sm−iSm−j+1 − Sm−i+1Sm−j )Sm−r+2) k −(Sm−iSm−j+2 − Sm−i+2Sm−j )Sm−r+1 + (Sm−i+1Sm−j+2 − Sm−i+2Sm−j+1)Sm−r) where {i, j, r} ⊆ {0, 1, . . . , m + 2} and i < j < r. Proof. (I) Define the polynomial 2 i−1 i+1 j−1 j j+1 m+1 1 c1 c1 . . . c1 c1 . . . c1 c1 c1 . . . c1 2 i−1 i+1 j−1 j j+1 m+1 1 c2 c . . . c c . . . c c c . . . c 2 2 2 2 2 2 2 ...... f(x) := ...... 1 c c2 . . . ci−1 ci+1 . . . cj−1 cj cj+1 . . . cm+1 m m m m m m m m 1 x x2 . . . xi−1 xi+1 . . . xj−1 xj xj+1 . . . xm+1 m P m−k k Then it is clear that f(x) = (a1x + a0) (−1) Sm−kx . The leading coefficient k=0 of f(x) is a1 and from Lemma 2.1, we have m Y a1 = det(Vi(c1, c2, . . . , cm)) = Sm−i (cj − ck). k Again from Lemma 2.1, we have det(B) = c1c2 . . . cm det(Vi−1(c1, c2, . . . , cm)) = m m Q Q SmSm−i+1 (cj − ck), thus a0 = Sm−i+1 (cj − ck). k (II) Define the polynomial f(x) = det(Vi,j(c1, c2, . . . , cm, x)). Then f(x) is a polyno- 212 The inverse matrix of Vandermonde matrix mial of degree less than or equal to m + 2 and it is clear that m 2 X m−k k f(x) = (a2x + a1x + a0) (−1) Sm−kx . k=0 The leading coefficient of f(x) is a2 and from part (I), we have m Y a2 = det(Vi,j(c1, c2, . . . , cm)) = (Sm−iSm−j+1 − Sm−i+1Sm−j) (cj − ck). k Now the coefficient of xj is zero, so m−j+2 m−j+1 m−j (−1) a2Sm−j+2 + (−1) a1Sm−j+1 + (−1) a0Sm−j = 0, and substituting the values of a2 and a0, yields m Y (cj − ck) (Sm−iSm−j+1 − Sm−i+1Sm−j)Sm−j+2 k −(Sm−iSm−j+2 − Sm−i+2Sm−j)Sm−r+1 +(Sm−i+1Sm−j+2 − Sm−i+2Sm−j+1)Sm−r . Remark 2.5. Lemmas 2.1 and 2.4 can be used to find the determinants and co- factor matrices of the matrices Vi(c1, . . . , cm) and Vi,j(c1, . . . , cm). Thus computing the inverses of these matrices is easy to determine, provided that Vi(c1, . . . , cm) and Vi,j(c1, . . . , cm) are invertible. However, finding the cofactor matrix as well as the inverse of the matrix Vi,j,r(c1, . . . , cm) are still needed to be evaluated. In view of E. A. Rawashdeh 213 Lemmas 2.1 and 2.4, it will be interesting to find the inverse of the matrix i1 i2 im c1 c1 . . . c1 i1 i2 im c2 c2 . . . c2 V (c , c , . . . , c ) = , i1,i2,...,im 1 2 m . . . ...... i1 i2 im cm cm . . . cm where {i1, i2, . . . , im} is an increasing sequence of non-negative integers, which we plan to discuss in a forthcoming paper. References [1] J.M. Corcuera, D. Nualart, W. Schoutens, Completion of a L´evymarket by power-jump assets, Finance Stochastics, 9(1) (2005), 109–127. [2] E. Rawashdeh, D. Mcdowell, L, Rakesh, The Stability of Collocation Methods for Higher-Order Volterra Integro-Differential Equations, IJMMS, 2005: 3075–3089 (2005). [3] A. Eisinberg, G. Fedele, On the inversion of the Vandermonde matrix, Appl. Math. Comput., 174 (2006), 1384–1397. [4] M. Goldwurm, V. 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