Metzler matrix

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

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

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 Metzler.

Definition and Terminology

Nonnegative matrices

Positive matrices

Delay differential equation

M-matrix

P-matrix

Z-matrix

Quasipositive-matrix

stochastic matrix

Properties

The exponential of a Metzler Matrix is a Nonnegative matrix because of the corresponding property for the exponential of a Nonnegative matrix.

A Metzler Matrix has an eigenvector in the nonnegative orthant with a nonnegative eigenvalue because of the corresponding property for nonnegative matrices.

Relevant Theorems

Perron–Frobenius theorem

See Also

Nonnegative matrices

Positive matrices

Delay differential equation

M-matrix

P-matrix

Z-matrix

Quasipositive-matrix

stochastic matrix

Bibliography

1. Berman, Abraham; Plemmons, Robert J. (1994). Nonnegative Matrices in the Mathematical Sciences. SIAM. ISBN 0-89871-321-8. {{cite book}}: Cite has empty unknown parameter: |1= (help)
2. Farina, Lorenzo; Rinaldi, Sergio (2000). Positive Linear Systems: Theory and Applications. New York: Wiley Interscience. {{cite book}}: Cite has empty unknown parameter: |1= (help)
3. Berman, Abraham; Neumann, Michael; Stern, Ronald (1989). Nonnegative Matrices in Dynamical Systems. Pure and Applied Mathematics. New York: Wiley Interscience. {{cite book}}: Cite has empty unknown parameter: |1= (help)
4. Kaczorek, Tadeusz (2002). Positive 1D and 2D Systems. London: Springer. {{cite book}}: Cite has empty unknown parameter: |1= (help)
5. Luenberger, David (1979). Introduction to Dynamic Systems: Theory, Modes & Applications. John Wiley & Sons. {{cite book}}: Cite has empty unknown parameter: |1= (help)