In mathematics, especially linear algebra, an M-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a Z-matrix) and whose eigenvalues have nonnegative real parts.
[4] These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability, and (4) semipositivity and diagonal dominance.
Let A be a n × n real Z-matrix, then the following statements are equivalent to A being a non-singular M-matrix: Positivity of principal minors Inverse-positivity and splittings Stability Semipositivity and diagonal dominance The primary contributions to M-matrix theory has mainly come from mathematicians and economists.
M-matrices arise naturally in some discretizations of differential operators, such as the Laplacian, and as such are well-studied in scientific computing.
Meanwhile, economists have studied M-matrices in connection with gross substitutability, stability of a general equilibrium and Leontief's input–output analysis in economic systems.