In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of -matrices, with every principal minor 0.
By a theorem of Kellogg, the eigenvalues of - and - matrices are bounded away from a wedge about the negative real axis as follows:
The class of nonsingular M-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and Z-matrices are nonsingular -matrices. The class of sufficient matrices is another generalization of -matrices.
The linear complementarity problem has a unique solution for every vector if and only if is a -matrix. This implies that if is a -matrix, then is a -matrix.
If the Jacobian of a function is a -matrix, then the function is injective on any rectangular region of .
A related class of interest, particularly with reference to stability, is that of -matrices, sometimes also referred to as -matrices. A matrix is a -matrix if and only if is a -matrix (similarly for -matrices). Since , the eigenvalues of these matrices are bounded away from the positive real axis.