A rank-reducing and division-free algorithm for inverse of square matrices

Abstract

The paper puts forward a new direct algorithm for computing the inverse of a square matrix. The algorithm adopts a skill to compute the inverse of a regular matrix via computing the inverse of another lower-ranked matrix and contains neither iterations nor divisions in its computations—it is division-free. Compared with other direct algorithms, the new algorithm is easier to implement with either a recursive procedure or a recurrent procedure and has a preferable time complexity for denser matrices. Mathematical deductions of the algorithm are presented in detail and analytic formulas are exhibited for time complexity and spatial complexity. Also, the recursive procedure and the recurrent procedure are demonstrated for the implementation, and applications are introduced with comparative studies to apply the algorithm to tridiagonal matrices and bordered tridiagonal matrices.