Metzler matrix

 a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero)

Metzler matrices appear in stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of Nonnegative matrices to matrices of the form M + aI where M is a Metzler matrix.

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  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,...,u_n} are the eigenvalues of an n-dimensional P-matrix, then

If {u₁,...,u_n}, , i = 1,...,n are the eigenvalues of an n-dimensional P₀-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 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)

Z-matrix

 the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, a Z-matrix Z satisfies

Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.

The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix.

Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingularM-matrices.

M-matrix

 An M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices).^[1]

A common characterization of an M-matrix is a non-singular square matrix with non-positive off-diagonal entries and all principal minors positive, but many equivalences are known. The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski.^[2]

A symmetric M-matrix is sometimes called a Stieltjes matrix.

M-matrices arise naturally in some discretizations of differential operators, particularly those with a minimum/maximum principle, such as the Laplacian, and as such are well-studied in scientific computing.

The LU factors of an M-matrix are guaranteed to exist and can be stably computed without need for numerical pivoting, also have positive diagonal entries and non-positive off-diagonal entries. Furthermore, this holds even for incomplete LU factorization, where entries in the factors are discarded during factorization, providing useful preconditioners for iterative solution.