In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of -matrices, with every principal minor 0.
Spectra of -matrices By a theorem of Kellogg, the eigenvalues of - and - matrices are bounded away from a wedge about the negative real axis as follows:
If are the eigenvalues of an -dimensional -matrix, where , then
If , , are the eigenvalues of an -dimensional -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 -matrix.
If the Jacobian of a function is a -matrix, then the function is injective on any rectangular region of .
A related class of interest, particularly with reference to stability, is that of -matrices, sometimes also referred to as -matrices. A matrix is a -matrix if and only if is a -matrix (similarly for -matrices). Since , 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 - and -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)