In mathematics, a -matrix is a complex square matrix with every principal minor-matrices, which are the closure of the class of 0.

Spectra of P-matrices

By a theorem of Kellogg, the eigenvalues of - matrices are bounded away from a wedge about the negative real axis as follows:

1. are the eigenvalues of an -dimensional , then

2. are the eigenvalues of an -matrix, then

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 has a unique solution for every vector if and only if is a -matrix. This implies that if is a -matrix, then is a Q-matrix.

If the Jacobian of a function is a .

-matrices. A matrix is a , the eigenvalues of these matrices are bounded away from the positive real axis.

See also

• Routh–Hurwitz matrix

• Linear complementarity problem

• M-matrix

• Q-matrix

• Z-matrix

• Perron–Frobenius theorem

Notes

References

• David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965)

• Li Fang, On the Spectra of -Matrices, Linear Algebra and its Applications 119:1-25 (1989)

• R. B. Kellogg, On complex eigenvalues of and matrices, Numer. Math. 19:170-175 (1972)