On integral \(EP_ r\) matrices
A square complex matrix A is called \(EP_ r\) matrix, if \(rank(A)=r\) and \(R(A)=R(A^*)\) where \(A^*\) denotes the conjugate transpose of A and R(A) denotes the range of A. \textit{R. Penrose} [Proc. Camb. Philos. Soc. 51, 406-413 (1955; Zbl 0065.246)] has proved that there exists a unique matrix \(A^+\) that satisfies the following equations \(AA^+A=A\), \(A^+AA^+=A^+\), \((AA^+)=AA^+\), \((A^+A)=A^+A\). A matrix X is called an \(\{\) \(\ell \}\)-inverse of A if \(AXA=A.\) The main results of this paper are the following: 1. Let A be an \(n\times n\) integral matrix. Then the following statements are equivalent: (i) A is \(EP_ r\) and \(A^ TA\) has an integral \(\{\) \(\ell \}\)-inverse. (ii) There exists a permutation matrix P such that \(PAP^ T=\left[ \begin{matrix} D\quad 0\\ 0\quad 0\end{matrix} \right]\), where D is an \(r\times r\) nonsingular matrix with \(| \det D| =1\). (iii) \(A=GLG^ T\), where L and G are integral, \(G^ TG=I_ r\) and L is \(r\times r\) nonsingular with \(| \det L| =1\). (iv) \(A^+\) is an integral and \(EP_ r\) matrix. (v) There exists a symmetric idempotent integral matrix E such that \(AE=EA\) and \(R(A)=R(E).\) 2. Let A and B be integral \(EP_ r\) matrices. Then, \(A^ TA\) has an integral \(\{\) \(\ell \}\)-inverse and \(R(A)=R(B)\) if and only if AB is \(EP_ r\) and \((AB)^+=B^+A^+\) is integral.
scientific article; zbMATH DE number 3886064
scientific article