Metzler matrix: Difference between revisions
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A '''Metzler Matrix''' is a [[matrix (mathematics)|matrix]] in which all the off-diagonal components are nonnegative (equal to or greater than zero) |
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In [[mathematics]], especially [[linear algebra]], a [[matrix (mathematics)|matrix]] is called '''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 quasipositive matrix is any matrix ''A'' which satisfies |
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: <math>\qquad \forall_{i\neq j}\, x_{ij} \geq 0.</math> |
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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 matrix | Nonnegative matrices]] to matrices of the form <math>M+aI</math> where M is Metzler. |
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== Definition and Terminology == |
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:<math>A=(a_{ij});\quad a_{ij}\geq 0, \quad i\neq j.</math> |
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[[Nonnegative matrices]] |
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Quasipositive matrices are also sometimes referred to as <math>Z^{(-)}</math>-matrices, as a [[Z-matrix (mathematics)|''Z''-matrix]] is equivalent to a negated quasipositive matrix. |
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[[Positive matrices]] |
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The [[Matrix exponential|exponential]] of a quasipositive matrix is a [[nonnegative matrix]]<ref name="Berman">Berman, Abraham and Robert Plemmons. Nonnegative Matrices in the Mathematical Sciences [http://books.google.com/books?id=MRB7SUc_u6YC&pg=PA146&lpg=PA146&dq=essentially+nonnegative&source=web&ots=_s6p4di-Vl&sig=q5KDuOHn3nCEjmgi4K2vCehGeBU&hl=en&sa=X&oi=book_result&resnum=7&ct=result#PPA146,M1].</ref>. |
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[[Delay differential equation]] |
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== References == |
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{{reflist}} |
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[[M-matrix]] |
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[[Category:Matrices]] |
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[[P-matrix]] |
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[[Z-matrix]] |
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[[Quasipositive-matrix]] |
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[[stochastic matrix]] |
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== Properties == |
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The [[Matrix exponential|exponential]] of a Metzler Matrix is a [[Nonnegative matrix]] because of the corresponding property for the exponential of a [[Nonnegative matrix]]. |
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A Metzler Matrix has an eigenvector in the nonnegative orthant with a nonnegative eigenvalue because of the corresponding property for nonnegative matrices. |
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== Relevant Theorems == |
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[[Perron–Frobenius theorem]] |
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== See Also == |
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[[Nonnegative matrices]] |
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[[Positive matrices]] |
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[[Delay differential equation]] |
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[[M-matrix]] |
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[[P-matrix]] |
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[[Z-matrix]] |
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[[Quasipositive-matrix]] |
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[[stochastic matrix]] |
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== Bibliography == |
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# {{cite book| |
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| last1 = Berman |
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| first1 = Abraham |
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| authorlink1=Abraham Berman |
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| last2 = Plemmons |
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| first2 = Robert J. |
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| authorlink2=Robert J. Plemmons |
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| title = Nonnegative Matrices in the Mathematical Sciences |
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| publisher = SIAM |
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| ISBN = 0-89871-321-8 |
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| year=1994}} |
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#{{cite book| |
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| last1 = Farina |
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| first1 = Lorenzo |
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| authorlink1=Farina Lorenzo |
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| last2 = Rinaldi |
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| first2 = Sergio |
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| authorlink2=Sergio Rinaldi |
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| title = Positive Linear Systems: Theory and Applications |
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| publisher = Wiley Interscience |
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| location= [[New York]] |
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| year=2000}} |
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#{{cite book| |
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| last1 = Berman |
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| first1 = Abraham |
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| authorlink1=Abraham Berman |
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| last2 = Neumann |
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| first2 = Michael |
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| authorlink2=Michael Neumann |
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| last3 = Stern |
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| first3 = Ronald |
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| authorlink3 = Ronald Stern |
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| title = Nonnegative Matrices in Dynamical Systems |
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| series = Pure and Applied Mathematics |
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| publisher = Wiley Interscience |
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| location= [[New York]] |
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| year=1989}} |
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#{{cite book| |
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| last1 = Kaczorek |
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| first1 = Tadeusz |
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| authorlink1=Tadeusz Kaczorek |
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| title = Positive 1D and 2D Systems |
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| publisher = Springer |
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| location= [[London]] |
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| year=2002}} |
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#{{cite book| |
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| last1 = Luenberger |
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| first1 = David |
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| authorlink1=David Luenberger |
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| title = Introduction to Dynamic Systems: Theory, Modes & Applications |
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| publisher = John Wiley & Sons |
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| year=1979}} |
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[[Category:Matrices]] |
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{{maths-stub}} |
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Revision as of 18:13, 11 January 2011
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 where M is Metzler.
Definition and Terminology
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
See Also
Bibliography
- 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) - Farina, Lorenzo; Rinaldi, Sergio (2000). Positive Linear Systems: Theory and Applications. New York: Wiley Interscience.
{{cite book}}: Cite has empty unknown parameter:|1=(help) - 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) - Kaczorek, Tadeusz (2002). Positive 1D and 2D Systems. London: Springer.
{{cite book}}: Cite has empty unknown parameter:|1=(help) - Luenberger, David (1979). Introduction to Dynamic Systems: Theory, Modes & Applications. John Wiley & Sons.
{{cite book}}: Cite has empty unknown parameter:|1=(help)
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