Complex square matrix for which every principal minor is positive
In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of formula_1-matrices, which are the closure of the class of P-matrices, with every principal minor formula_2 0.
Spectra of P-matrices.
By a theorem of Kellogg, the eigenvalues of P- and formula_1- matrices are bounded away from a wedge about the negative real axis as follows:
If formula_4 are the eigenvalues of an n-dimensional P-matrix, where formula_5, then
formula_6
If formula_4, formula_8, formula_9 are the eigenvalues of an n-dimensional formula_1-matrix, then
formula_11
Remarks.
The class of nonsingular "M"-matrices is a subset of the class of P-matrices. More precisely, all matrices that are both P-matrices and "Z"-matrices are nonsingular M-matrices. The class of sufficient matrices is another generalization of P-matrices.
The linear complementarity problem formula_12 has a unique solution for every vector q if and only if M is a P-matrix. This implies that if M is a P-matrix, then M is a Q-matrix.
If the Jacobian of a function is a P-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 A is a formula_14-matrix if and only if formula_17 is a P-matrix (similarly for formula_1-matrices). Since formula_19, the eigenvalues of these matrices are bounded away from the positive real axis.