Tridiagonal matrix algorithm

Abstract: The closed form inverse of a tridiagonal matrix, which is a slight generalization of a matrix considered by D. Kershaw (Math. Comp., v. 23, 1969, pp. 189-191), is given in this note. If the matrix has integer elements, an integer multiple of the inverse can be computed by integer arithmetic, that is, without machine roundoff error. Forms of B and B~l. Let B = (/>,,) be an n X n matrix given by ba = bi, i = j, (1) = «,.,-,. i < j, = í.-i.í. i > j, where b¡ = /3„_i+1 and Bti is the Kronecker delta. Define rk = —(&*/• *_i + rt_2), k = 2, ■ ■ ■ , n — 1, and r = (bnrn_l + rn_2), where r0 = 1 and r, = —bt. Let C = (cit) be an n X n matrix, where i2, Cij = /•''/•,-!/•„_,, i á /, = c,i, » > ;, then C = B1. An elementary but lengthy proof of this is to show that BC is an n X n identity matrix. It can also be shown that r = (—l)n+1 det (B). This implies that ri_1r,_, = (— l)n+1/l„, where AH is the cofactor of ft,-,-. If bu i = 1, • • • , n, is an integer, then rk,k = 2, ■ ■ ■ ,n — \\, and r are integers. This implies that rc^ = /•,_1r„_, is an integer, that is, an integer multiple of B'1 can be generated by integer arithmetic. Similarly, if one had a linear system to solve, which was represented by a matrix of the form (1) with integer elements, then an integer multiple of the solution could be computed by integer arithmetic if the known vector had rational coordinates. Generalization to Partitioned Matrices. If the elements b¡, i = 1, • • • , n, 1, 0 of B are replaced by m X m matrices B¡, i = 1, • • • , n, I, 0, where lis the identity matrix and 0 is the null matrix, respectively, then the inverse of B is given by the partitioned matrix (2) if BiBj = 5,5,, i, j = 1, • • • , n. Here r_I is considered the inverse of some m X m matrix. Applied Mathematics Division ARDC, BRL Aberdeen Proving Ground, Maryland 21005 Received November 26, 1969. AMS Subject Classifications. Primary 1510, 1515; Secondary 1079, 1548.