Analysis for M-Matrices

The positive definiteness A > 0 establishes a partial-order relation in the algebra of the Hermitian matrices. Another order relation in the algebra of all real matrices is defined by elementwise inequalities: 6.1.1a$$ A\, > \,B\,: \Leftrightarrow \,{a_{\alpha \beta }}\, > \,{b_{\alpha \beta }}\,\,for\,all\,\alpha ,\,\beta \, \in \,I, $$6.1.1b$$ A\; \ge \;B: \Leftrightarrow {a_{\alpha \beta }}\; \ge \;{b_{\alpha \beta }}\;\;{\rm{for all }}\alpha {\rm{,}}\;\beta \; \in \;I. $$ In this chapter, we use the signs «>» and «≥» only in the componentwise sense (1a, b).