Infimum of a pair of matrices
For a given matrix A, the author denotes by A - a generalized inverse of A, i.e. a matrix G satisfying \(AGA=A\). If rank \(A=rank\) A 2, then the group inverse \(A^{\#}\) of A is the unique solution G of the system of matrix equations \(AGA=A\), \(GAG=G\), \(AG=GA\). In the set of all matrices of fixed dimension, the binary relation \(A\leq B\) whenever A \(-A=A\)-B and AA \(-=BA\)-, is a partial order, called the minus partial order. In the set of all n by n matrices A satisfying rank \(A=rank\) A 2, the binary relation \(A^{\#}<B\) whenever \(A^{\#}A=A^{\#}B\) and \(AA^{\#}=BA^{\#}\), is a partial order, called the sharp order. The author investigates when an infimum of A and B with respect to each order exists. For this, he obtains a criterion for the possibility to extend a given k by m matrix \((k<m)\) to an idempotent m by m matrix, as well as other, similar, results on matrix idempotents.
scientific article; zbMATH DE number 4059587
scientific article