a

**is a complex square matrix with every principal minor > 0. A closely related class is that of***P*-matrix*P*_{0}-matrices, which are the closure of the class of*P*-matrices, with every principal minor 0.

Spectra of *P*-matrices

By a theorem of Kellogg, the eigenvalues of

*P*– and*P*_{0}– matrices are bounded away from a wedge about the negative real axis as follows:- If {
*u*_{1},…,*u*_{n}} are the eigenvalues of an*n*-dimensional*P*-matrix, then - If {
*u*_{1},…,*u*_{n}}, ,*i*= 1,…,*n*are the eigenvalues of an*n*-dimensional*P*_{0}-matrix, then

## Notes

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.If the Jacobian of a function is a

*P*-matrix, then the function is injective on any rectangular region of .A related class of interest, particularly with reference to stability, is that of

*P*^{( − )}-matrices, sometimes also referred to as*N*−*P*-matrices. A matrix*A*is a*P*^{( − )}-matrix if and only if ( −*A*) is a*P*-matrix (similarly for*P*_{0}-matrices). Since σ(*A*) = − σ( −*A*), the eigenvalues of these matrices are bounded away from the**positive**real axis.

References

- R. B. Kellogg, On complex eigenvalues of
*M*and*P*matrices,*Numer. Math.*19:170-175 (1972) - Li Fang, On the Spectra of
*P*– and*P*_{0}-Matrices,*Linear Algebra and its Applications*119:1-25 (1989) - D. Gale and H. Nikaido, The Jacobian matrix and global univalence of mappings,
*Math. Ann.*159:81-93 (1965)