A PRACTICAL METHOD FOR MATRIX INVERSION

Abstract

This report presents a matrix inversion method that has the following features: general applicability for any non-singular (n × n) matrix A with either real or complex elements a_ij , i, j = 1, 2, ..., n; univocally defined matrix operations; analytical representation of the sought-for inverse matrix A⁻¹ as a product of three uniquely specified non-singular triangular matrices (A⁻¹ = PGV, P and V lower triangular, G upper triangular); and convenient (minimal) number, n³, of required multiplication/division operations.

The inversion procedure is carried out in two stages: I) transformation of matrix A into an upper triangular matrix T having unit diagonal elements; II) transformation of matrix T into the (n × n) unit matrix U⁽ⁿ⁾ having elements \(u_{ij}^{(n)}\) = 0 if i ≠ j, and = 1 if i = j.

The first stage conforms with a Gaussian elimination procedure that can be carried out in the natural order, through n consecutive ordered steps, since at each step univocally prescribed non-zero diagonal elements (leading pivots) are made available.  A sequence of transformed non-singular (n × n) matrices {A^(k)}, k = 1, 2, ..., n, having elements \(a_{ij}^{(k)}\), is obtained with pivots \(a_{kk}^{(k)}\) = 1.  The viability of the procedure is assured by the ordered use of univocally defined, very simple non-singular lower triangular (n × n) operational matrices P^(k) which post-multiply A^(k−1) (A⁽⁰⁾ coinciding with A) and transform diagonal element \(a_kk^{(k-1)}\) into a leading pivot having a positive non-zero value given by the sum of the absolute values of all the elements in its row.

A simple numerical example is detailed for illustrating practical aspects.

The present matrix inversion method is free from any operational ambiguity.

The simplicity and univocal definiteness of its transformations are expected to provide operational advantages for the development of related numerical algorithms both for finding matrix inverses and for solving systems of linear algebraic equations.  Further useful features are related with the final triangular matrix factorization achieved which, in particular, allows an immediate computation of the determinants of matrices A and A⁻¹.