Quasipositive matrix
Encyclopedia
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.
, 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.
of a Metzler (or quasipositive) 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.
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
in which all the off-diagonal components are nonnegative (equal to or greater than zero)
It is named after the American economist Lloyd Metzler
Lloyd Metzler
Lloyd Appleton Metzler was an American economist best known for his contributions to international trade theory. He was born at Lost Springs, Kansas in 1913. Although most of his career was spent at the University of Chicago, he was not a member of the Chicago school, but rather a Keynesian...
.
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
Nonnegative matrix
A nonnegative matrix is a matrix in which all the elements are equal to or greater than zeroA positive matrix is a matrix in which all the elements are greater than zero...
to matrices of the form M + aI where M is a Metzler matrix.
Definition and terminology
In mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
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 equationDelay differential equationIn mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
- M-matrixM-matrixIn mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices In mathematics, especially linear algebra, an M-matrix is a Z-matrix with...
- P-matrix
- Z-matrix
- stochastic matrixStochastic matrixIn mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science...
Properties
The exponentialMatrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
of a Metzler (or quasipositive) matrix is a nonnegative matrix
Nonnegative matrix
A nonnegative matrix is a matrix in which all the elements are equal to or greater than zeroA positive matrix is a matrix in which all the elements are greater than zero...
because of the corresponding property for the exponential of a Nonnegative matrix
Nonnegative matrix
A nonnegative matrix is a matrix in which all the elements are equal to or greater than zeroA positive matrix is a matrix in which all the elements are greater than zero...
.
A Metzler matrix has an eigenvector in the nonnegative orthant with a nonnegative eigenvalue because of the corresponding property for nonnegative matrices.
See also
- Nonnegative matrices
- Delay differential equationDelay differential equationIn mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
- M-matrixM-matrixIn mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices In mathematics, especially linear algebra, an M-matrix is a Z-matrix with...
- P-matrix
- Z-matrix
- Quasipositive-matrix
- stochastic matrixStochastic matrixIn mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science...