|
|
| Line 1: |
Line 1: |
| 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)
| | The writer's name is Christy. What me and my family adore is to climb but I'm thinking on beginning something new. Credit authorising is how she tends to make a living. Her family lives in Ohio.<br><br>my website ... [http://brazil.amor-amore.com/irboothe psychic readings online] |
| | |
| : <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}}
| |
The writer's name is Christy. What me and my family adore is to climb but I'm thinking on beginning something new. Credit authorising is how she tends to make a living. Her family lives in Ohio.
my website ... psychic readings online