Z matrix (math)

In mathematics, a Z-matrix is a matrix in which the entries outside the main diagonal are less than or equal to zero. A Z-matrix therefore satisfies for all . According to this definition, a Z matrix is ​​exactly the negative of a Metzler matrix . The latter are also referred to as quasi-positive matrices, which is why the term quasi-negative matrix also appears in the literature for Z-matrices , albeit rarely and usually only in a context in which quasi-positive matrices are referred to. ${\displaystyle Z=(z_{ij})}$ ${\displaystyle z_{ij}\leq 0}$ ${\displaystyle i\neq j}$

literature

• Miroslav Fiedler, Vlastimil Pták: On matrices with non-positive off-diagonal elements and positive principal minors. In: Czechoslovak Mathematical Journal No. 12/1962, pp. 382-400.
• Miroslav Fiedler, Vlastimil Pták: Some generalizations of positive definiteness and monotonicity. In: Numerical Mathematics No. 9/1966, pp. 163–172.
• Abraham Berman, Robert J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York 1979.