A determinant for rectangular matrices

Abstract

The familiar notion of the determinant is generalised to include rectangular matrices. An expression for a normalised generalised inverse of a matrix is given in terms of its determinant and a possible generalisation of the Schur complement is discussed as a simple application.

Impulse Delay in the Cardiac Conduction System

“…The superposition of the sum of the responses to all of the component weighted pulses in (20) is given by…”

“…consisting of triple matrices P, Q, R assumed to be positive definite, and where in the present case A 0 and A 1 are Moore-Penrose invertible [18,19,20]). Therefore:…”

A Weak Generalized Inverse Applied to Redundancy Solving of Serial Chain Robots

“…Radic [7] proposed at his time a first definition of the determinant of such rectangular matrices. We will however prefer a more recent definition proposed by Joshi [6] for which it is easy to show that it is equivalent to the simple form:…”

“…We will consider in this paper a right inverse of Mi.e. a matrix X verifying MX=I m and as a consequence a {1,2,3}-Inverseintroduced in the 80' by the Indian mathematician Joshi [6] in a work first devoted to determinant definition of non-rectangular matrices. In section 2, we introduce Joshi's determinant that we relate to zonotope theory and then Joshi's weak generalized inverse for which we propose an associated symbolic expression of the projection operator.…”

Cullis-Radić determinant of a rectangular matrix which has a number of identical columns

“…Independently of Cullis, in 1966, Radić [11] proposed the following definition of the determinant, which turns out to be equivalent to the Cullis's definition (the equivalence follows from [4, §30]). It is worth noting here that there are also other definitions of the determinant of a rectangular matrix which are not equivalent to the definitions of Cullis and Radić, see for example [2,3,5,9,18,19,21].…”