P-matrix

In mathematics, a -matrix is a complex square matrix with every principal minor > 0. A closely related class is that of formula_1-matrices, which are the closure of the class of -matrices, with every principal minor formula_2 0.
Spectra of -matrices.
By a theorem of Kellogg, the eigenvalues of - and formula_1- matrices are bounded away from a wedge about the negative real axis as follows:
Remarks.
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 formula_12 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 formula_13.
A related class of interest, particularly with reference to stability, is that of formula_14-matrices, sometimes also referred to as formula_15-matrices. A matrix is a formula_14-matrix if and only if formula_17 is a -matrix (similarly for formula_1-matrices). Since formula_19, the eigenvalues of these matrices are bounded away from the positive real axis.