P-matrix

a P-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of P0-matrices, which are the closure of the class of P-matrices, with every principal minor geq 0.


Spectra of P-matrices

By a theorem of Kellogg, the eigenvalues of P– and P0– matrices are bounded away from a wedge about the negative real axis as follows:
If {u1,…,un} are the eigenvalues of an n-dimensional P-matrix, then

|arg(u_i)| < pi - frac{pi}{n}, i = 1,...,n
If {u1,…,un}, u_i neq 0, i = 1,…,n are the eigenvalues of an n-dimensional P0-matrix, then

|arg(u_i)| leq pi - frac{pi}{n}, i = 1,...,n

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 mathbb{R}^n.
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 P0-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 P0-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)