Andrica's conjecture: Difference between revisions
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In [[mathematics]], a '''Metzler matrix''' is a [[matrix (mathematics)|matrix]] in which all the off-diagonal components are nonnegative (equal to or greater than zero) | |||
: <math>\qquad \forall_{i\neq j}\, x_{ij} \geq 0.</math> | |||
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 matrix | 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 (mathematics)|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 | |||
:<math>A=(a_{ij});\quad a_{ij}\geq 0, \quad i\neq j.</math> | |||
Metzler 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. | |||
* [[Nonnegative matrices]] | |||
* [[Positive matrix]] | |||
* [[Delay differential equation]] | |||
* [[M-matrix]] | |||
* [[P-matrix]] | |||
* [[Z-matrix (mathematics)|Z-matrix]] | |||
* [[Stochastic matrix]] | |||
== Properties == | |||
The [[Matrix exponential|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]] | |||
== See also == | |||
* [[Nonnegative matrices]] | |||
* [[Delay differential equation]] | |||
* [[M-matrix]] | |||
* [[P-matrix]] | |||
* [[Z-matrix (mathematics)|Z-matrix]] | |||
* [[Quasipositive-matrix]] | |||
* [[Stochastic matrix]] | |||
== Bibliography == | |||
{{reflist}} | |||
*{{cite book| | |||
| last1 = Berman | |||
| first1 = Abraham | |||
| authorlink1=Abraham Berman | |||
| last2 = Plemmons | |||
| first2 = Robert J. | |||
| authorlink2=Robert J. Plemmons | |||
| title = Nonnegative Matrices in the Mathematical Sciences | |||
| publisher = SIAM | |||
| ISBN = 0-89871-321-8 | |||
| year=1994}} | |||
*{{cite book| | |||
| last1 = Farina | |||
| first1 = Lorenzo | |||
| authorlink1=Farina Lorenzo | |||
| last2 = Rinaldi | |||
| first2 = Sergio | |||
| authorlink2=Sergio Rinaldi | |||
| title = Positive Linear Systems: Theory and Applications | |||
| publisher = Wiley Interscience | |||
| location= [[New York]] | |||
| year=2000}} | |||
*{{cite book| | |||
| last1 = Berman | |||
| first1 = Abraham | |||
| authorlink1=Abraham Berman | |||
| last2 = Neumann | |||
| first2 = Michael | |||
| authorlink2=Michael Neumann | |||
| last3 = Stern | |||
| first3 = Ronald | |||
| authorlink3 = Ronald Stern | |||
| title = Nonnegative Matrices in Dynamical Systems | |||
| series = Pure and Applied Mathematics | |||
| publisher = Wiley Interscience | |||
| location= [[New York]] | |||
| year=1989}} | |||
*{{cite book| | |||
| last1 = Kaczorek | |||
| first1 = Tadeusz | |||
| authorlink1=Tadeusz Kaczorek | |||
| title = Positive 1D and 2D Systems | |||
| publisher = Springer | |||
| location= [[London]] | |||
| year=2002}} | |||
*{{cite book| | |||
| last1 = Luenberger | |||
| first1 = David | |||
| authorlink1=David Luenberger | |||
| title = Introduction to Dynamic Systems: Theory, Modes & Applications | |||
| publisher = John Wiley & Sons | |||
| year=1979}} | |||
[[Category:Matrices]] | |||
{{Linear-algebra-stub}} | |||
Revision as of 15:45, 24 November 2013
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero)
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
Metzler matrices are also sometimes referred to as -matrices, as a Z-matrix is equivalent to a negated quasipositive matrix.
- Nonnegative matrices
- Positive matrix
- Delay differential equation
- M-matrix
- P-matrix
- Z-matrix
- Stochastic 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
See also
- Nonnegative matrices
- Delay differential equation
- M-matrix
- P-matrix
- Z-matrix
- Quasipositive-matrix
- Stochastic matrix
Bibliography
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