Moore-Penrose inverse of matrices on idempotent semirings (Q2706279)

The author gives necessary and sufficient conditions on a matrix \(A\) over a commutative, algebraically complete semiring in which the cancellation and stabilization condition holds and in which \(\oplus \) is idempotent and the partial ordering ``\(\leq\)'' induced by \(\oplus\) is total so that it admits a Moore-Penrose inverse. The main result is the following: The three conditions (i) \(A^{+}\), the Moore-Penrose inverse of \(A\), exists, (ii) there exist presentation matrices \(P\),\(Q\) such that NEWLINE\[NEWLINEPAQ=\left( \begin{matrix} F_{1} & 0 & \cdots & 0 & 0 \\ 0 & F_{2} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & F_{k} & 0 \\ 0 & 0 & \cdots & 0 & 0 \end{matrix}\right)NEWLINE\]NEWLINE where the blocks \(F_{i}\) are positive but not necessarily square, and (iii) there exists an invertible diagonal matrix \(D\) such that \( A^{+}=DA^{t}\), are equivalent. NEWLINENEWLINENEWLINEOne motivation for studying the Moore-Penrose inverse over idempotent semirings is the fact that the idempotent semiring \(\mathbb{R},\max ,\times \) has applications in areas like optimal control and discrete event systems. The paper also provides a program in MATLAB to compute the Moore-Penrose inverse of a matrix and compares similar results on matrices over \((\mathbb{R},+,\times)\).