An Algorithm for Constructing Random Inverses of Non-Square Matrices
Across Arbitrary Fields
Abstract
In the realm of linear algebra, the notion of matrix inversion plays a
crucial role. While the inversion of square matrices is well-known and
results in a unique inverse, however, the non-square inverse matrice is
not unique and in fact, the number of inverses for a non-square matrix
can be as vast as q^m(n−m), where q signifies the order of the
underlying field. In this paper, we embark on a journey to construct
these elusive inverse matrices, harnessing the power of arbitrary
fields. Arbitrary fields, including prime fields, finite fields, real
fields, and complex fields. These fields find practical applications
that are essential to contemporary technology. I have written 5 MATLAB
programs that able to construct random inverses in different fields
based on the given algorithm.