Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant

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1 Advances in Numerical Analysis Volume 2011, Article ID , 6 pages doi: /2011/ Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant F. N. Valvi Department of Mathematics, University of Patras, Patras, Greece Correspondence should be addressed to F. N. Valvi, fvalvi@upatras.gr Received 29 December 2010; Revised 3 October 2011; Accepted 10 October 2011 Academic Editor: Alfredo Bermudez De Castro Copyright q 2011 F. N. Valvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this paper is to present the structure of a class of matrices that enables explicit inverse to be obtained. Starting from an already known class of matrices, we construct a Hadamard product that derives the class under consideration. The latter are defined by 4n 2 parameters, analytic expressions of which provide the elements of the lower Hessenberg form inverse. Recursion formulae of these expressions reduce the arithmetic operations in evaluating the inverse to O n Introduction In 1, a class of matrices K n a ij with elements 1, i j, a ij a j, i > j 1.1 is treated. A generalization of this class is presented in 2 by the matrix G n b ij, where b j, i j, b ij a j, i > j. 1.2 In this paper, we consider a more extended class of matrices, M, and we deduce in analytic form its inverse and determinant. The class under consideration is defined by the Hadamard product of G n and a matrix L, which results from G n first by assigning the values

2 2 Advances in Numerical Analysis a i l n i 1 and b i k n i 1 to the latter in order to get a matrix K, say, and then by the relation L PK T P, where P p ij is the permutation matrix with elements 1, i n j 1, p ij 0, otherwise. 1.3 The so-constructed class is defined by 4n 2 parameters, and its inverse has a lower Hessenberg analytic expression. By assigning particular values to these parameters, a great variety of test matrices occur. It is worth noting that the classes L and G n that produce the class M L G n belong to the extended DIM classes presented in 3 as well as to the categories of the upper and lower Brownian matrices, respectively, as they have been defined in The Class of Matrices and Its Inverse Let M m ij be the matrix with elements k i b j, i j, m ij l i a j, i > j, 2.1 that is, k 1 b 1 k 1 b 2 k 1 b 3 k 1 b n 1 k 1 b n l 2 a 1 b 2 b 3 b n 1 b n l 3 a 1 l 3 a 2 k 3 b 3 k 3 b n 1 k 3 b n M. 2.2 l n 1 a 1 l n 1 a 2 l n 1 a 3 k n 1 b n 1 k n 1 b n l n a 1 l n a 2 l n a 3 l n a n 1 k n b n If M 1 μ ij is its inverse, then the following expressions give its elements k i 1 b i 1 l i 1 a i 1, c i 1 c i i j 2, 3,...,n 1,, c 0 c 1 i j 1, b n 1, i j n, μ ij c n 1 c n 1 d i 1 j 1g i j i ν j 1 f ν i ν j 1 c, i > j, ν 2.3 1, i j 1, c i 0, i < j 1,

3 Advances in Numerical Analysis 3 where c i k i 1 b i l i 1 a i, i 1, 2,...,n 1, c 0 k 1, c n b n, d i a i 1 b i a i b i 1, i 1, 2,...,n 2, d 0 a 1, f i l i a i k i b i, i 2, 3,...,n 1, 2.4 g i k i 1 l i k i l i 1, i 2, 3,...,n 1, g n l n, with i 1 ν j 1 f ν 1 whenever i j 1, 2.5 and with the obvious assumption c i / 0, i 0, 1, 2,...,n The Proof We prove that the expressions 2.3 give the inverse matrix M 1. To that purpose, we reduce M to the identity matrix by applying elementary row operations. Then the product of the corresponding elementary matrices gives the inverse matrix. In particular, adopting the conventions 2.4, we apply the following sequence of row operations: Operation 1. row i k i /k i 1 row i 1, i 1, 2,...,n 1, which gives the lower triangular matrix k 1 c 1 0 a 1 g 2 c 2 k 3 k a 1 a 2 k n 1 g n 1 g n 1 c n 1 0 k n k n k n g n a 1 g n a 2 g n a n 1 g n b n Operation 2. row i k i g i /k i 1 g i 1 row i 1, i n, n 1,...,3, k n 1 1, which results in a bidiagonal matrix with main diagonal ( k1 c 1, c 2 k 3,..., k ) n 1c n 1,k n b n k n 3.2

4 4 Advances in Numerical Analysis and lower first diagonal ( a1 g 2 k 3, k 3g 3 f 2 k 4 g 2,..., k n 1g n 1 f n 2, k ) ng n f n k n g n 2 g n 1 Operation 3. row 2 a 1 g 2 /k 1 k 3 c 1 row 1, and row i k i k i g i f i 1 /k i 1 k i 1 g i 1 c i 1 row i 1, i 3, 4,...,n, which gives the diagonal matrix k1 c 1 c 2 k 3 k n 1 c n 1 k n k n b n. 3.4 Operation 4. k i 1 /k i c i row i, i 1, 2,...,n, which gives the identity matrix. Operations 1 4 transform the identity matrix to the following forms, respectively: Form 1. The upper bidiagonal matrix consisting of the main diagonal 1, 1,...,1 3.5 and the upper first diagonal ( k 1,,..., k ) n k 3 k n Form 2. The tridiagonal matrix 1 k k 3 0 k 3g 3 k 3 k 4 l 2 l 4 k 4 g 2 k 4 g k n 1 k n l n 2 k n 2 l n k n 1 k n g n 2 k n k ng n k n l n 1 g n 1 g n 1

5 Advances in Numerical Analysis 5 Form 3. The lower Hessenberg matrix s a 1g n 1 k n 1 f 2 f n 2 k 1 k n c 1 c n 2 s a 1g n k n f 2 f n 1 k 1 c 1 c n 1 1 k 1 a 1g 2 k 3 b 1 l 3 a 1 k 1 k 3 c 1 k 3 c 1 a 1 g 3 k 3 f 2 d 1g 3 k 3 k 1 k 4 c 1 c 2 k 4 c 1 c 2 s d 1g n 1 k n 1 f 3 f n 2 k n c 1 c 2 c n 2 k n 1 k n c n 2 k n b n 2 l n a n 2 s d 1g n k n f 3 f n 1 c 1 c 2 c n 1 d n 2g n k n c n 2 c n 1 k n 1 k n b n 1 k n c n 1, 3.8 where the symbol s stands for the quantity 1 i j. Form 4. The matrix whose elements are given by the expressions 2.4. The determinant of M takes the form det M k 1 b n b 1 l 2 a 1 k n b n 1 l n a n Evidently, M is singular if c i 0 for some i {0, 1, 2,...,n}. 4. Numerical Complexity The inverse of the matrix M is given explicitly by the expressions 2.3. However, a careful reader could easily derive the recursive algorithm that gives the elements under the main diagonal of M 1. In particular, μ i,i s 1 d i 2g i μ i,i 1, c i 2 c i 1 c i i 2, 3,...,n, d i s 2f i s d i s 1 c i s 2 μ i,i s, i 3, 4,...,n, s 1, 2,...,i 2, 4.1 or, alternatively, μ j s 1,j μ j 1,j d j 1g j 1 c j 1 c j c j 1, j 1, 2,...,n 1, g j s 1f j s g j s c j s 1 μ j s,j, j 1, 2,...,n 2, s 1, 2,...,n j 1, 4.2 where the c i, d i, f i,andg i are given by the relations 2.4. By use of the above algorithms, the estimation of the whole inverse of the matrix M is carried out in 2n 2 11n 19

6 6 Advances in Numerical Analysis multiplications/divisions, since the coefficient of μ ij depends only on the second first subscript, respectively, and in 5n 9 additions/subtractions. 5. Remarks When replacing k i, b i,anda i by a i, k i,andb i, respectively, the matrix M see 2.2 is transformed into the transpose C T of the matrix C 5,Section2. However, the primary fact for a test matrix is the structure of its particular pattern, which succeeds in yielding the analytic expression of its inverse. In the present case, the determinants of the minors of the elements m ij,i>j 1, vanish to provide the lower Hessenberg type inverse. In detail, each minor of M, that occurs after having removed the ith row and jth column, i>j 1, has the determinant of its i 1 i 1 upper left minor equal to zero, since the last two columns of the latter are linearly dependent. Accordingly, by using induction, it can be proved that all the remaining upper left minors of order i, i 1,...,n 1 vanish. References 1 H. W. Milnes, A note concerning the properties of a certain class of test matrices, Mathematics of Computation, vol. 22, pp , R. J. Herbold, A generalization of a class of test matrices, Mathematics of Computation, vol. 23, pp , G. Carayannis, N. Kalouptsidis, and D. G. Manolakis, Fast recursive algorithms for a class of linear equations, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 30, no. 2, pp , M. J. C. Gover and S. Barnett, Brownian matrices: properties and extensions, International Systems Science, vol. 17, no. 2, pp , F. N. Valvi, Analytic inversion for a general case of certain classes of matrices, International Computer Mathematics, vol. 72, no. 2, pp , 1999.

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