- Upper and lower bounds for ranks of matrix products composed by generalized inverses with applications Yongge Tian(yongge.tiangmail.com) Bo Jiang(jiangboliumengyugmail.com) Abstract: In matrix theory and its applications, we often meet with various matrix expressions or matrix equalities that involve inverses and generalized inverses of matrices, one of the simplest forms of such matrix expressions is given by \$P^{(i,\ldots,j)}NQ^{(i,\ldots,j)}\$, where \$P\$, \$Q\$, and \$N\$ are three given matrices, \$P^{(i,\ldots,j)}\$ and \$Q^{(i,\ldots,j)}\$ are the eight frequently used \$\{i,\ldots, j\}\$ generalized inverses of \$P\$ and \$Q\$. This paper presents a general approach by using the matrix rank methodology. We first establish exact formulas for calculating the maximum and minimum ranks of \$P^{(i,\ldots,j)}NQ^{(i,\ldots,j)}\$ with respect to \$\{i,\ldots, j\}\$-inverses of \$P\$ and \$Q\$, and then use the formulas to characterize a variety of algebraic properties of these products. Some applications to the special products \$A^{(i,\ldots,j)}(A+B)B^{(i,\ldots,j)}\$ and \$(BC)^{(i,\ldots,j)}B(AB)^{(i,\ldots,j)}\$ are also presented. Keywords: 15A03; 15A09; 15A24 Category 1: Global Optimization Citation: Download: [PDF]Entry Submitted: 03/09/2017Entry Accepted: 03/10/2017Entry Last Modified: 03/09/2017Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.