An integer arithmetic method to compute generalized matrix inverse and solve linear equations exactly

Whether a matrix A over a complex field is singular square, or rectangular, it has always a generalized inverse (g-inverse) over the (complex) field. The true inverse exists only when ,4 is nonsingular (i.e., a square matrix whose determinant is not zero). However, a g-inverse of an m x n matrix of rank r involves considerable errors if the rth order submatrices are near-singular. Further, the rank shown by the g-inverse may be less than the actual rank. In fact, identical are the pitfalls when a (square) near-singular matrix is inverted. We present here a method that uses integer arithmetic to (i) transform an m x n integral matrix to a Smith Diagonal Form (defined later) without requiring to compute the greatest common divisors (GCDs) of the matrix elements as required in computing certain g-inverses (Hurt and Waid [4], Ben-Israel and Greville [2]), (ii) compute a reflexive g-inverse (Bowman and Burdet [3], Ben-Israel and Greville [2], Krishnamurthy and Sen [5]), (iii) obtain a solution vector x of Ax=b, b being 0 (null column vector) or not. Since any computing system can represent only the rational numbers, we can, without any loss of generality, assume the inputs (here the matrix A and the right-hand-side column vector b) integral.