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

$${\displaystyle \forall _{i\neq j}\,x_{ij}\geq 0.}$$

It is named after the American economist Lloyd Metzler.

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.

Definition and terminology

In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies

$$A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.$$

Metzler matrices are also sometimes referred to as $$Z^{{(-)}}$$ -matrices, as a Z-matrix is equivalent to a negated quasipositive matrix.
Properties

The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time finite-state Markov processes are always Metzler matrices, and that probability distributions are always non-negative.

A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.
Relevant theorems

Perron–Frobenius theorem

Nonnegative matrices
Delay differential equation
M-matrix
P-matrix
Z-matrix
Hurwitz matrix
Stochastic matrix
Positive systems

Bibliography

Berman, Abraham; Plemmons, Robert J. (1994). Nonnegative Matrices in the Mathematical Sciences. SIAM. ISBN 0-89871-321-8.
Farina, Lorenzo; Rinaldi, Sergio (2000). Positive Linear Systems: Theory and Applications. New York: Wiley Interscience.
Berman, Abraham; Neumann, Michael; Stern, Ronald (1989). Nonnegative Matrices in Dynamical Systems. Pure and Applied Mathematics. New York: Wiley Interscience.
Kaczorek, Tadeusz (2002). Positive 1D and 2D Systems. London: Springer.
Luenberger, David (1979). Introduction to Dynamic Systems: Theory, Modes & Applications. John Wiley & Sons. pp. 204–206. ISBN 0-471-02594-1.
Kemp, Murray C.; Kimura, Yoshio (1978). Introduction to Mathematical Economics. New York: Springer. pp. 102–114. ISBN 0-387-90304-6.

Matrix classes
Explicitly constrained entries

(0,1) Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Binary Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Markov Metzler Monomial Moore Nonnegative Partitioned Parisi Pentadiagonal Permutation Persymmetric Polynomial Positive Quaternionic Sign Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Unitary Vandermonde Walsh Z

Constant

Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero

Conditions on eigenvalues or eigenvectors

Companion Convergent Defective Diagonalizable Hurwitz Positive-definite Stability Stieltjes

Satisfying conditions on products or inverses

Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing

With specific applications

Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental (linear differential equation) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X–Y–Z

Used in statistics

Bernoulli Centering Correlation Covariance Design Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition

Used in graph theory

Used in science and engineering

Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)

Related terms

Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian

List of matrices Category:Matrices