Andrica's conjecture

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

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

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.

• 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

• Perron–Frobenius theorem

See also

• Nonnegative matrices
• Delay differential equation
• M-matrix
• P-matrix
• Z-matrix
• Quasipositive-matrix
• Stochastic matrix

Bibliography

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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