In
mathematics, a Metzler matrix is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 Appleton Metzler (1913 – 26 October 1980) was an American economist best known for his contributions to international trade theory. He was born in Lost Springs, Kansas in 1913. Although most of his career was spent at the University of ...
.
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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
, a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
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.
Properties
The
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
of a Metzler (or quasipositive) matrix is a
nonnegative matrix
In mathematics, a nonnegative matrix, written
: \mathbf \geq 0,
is a matrix in which all the elements are equal to or greater than zero, that is,
: x_ \geq 0\qquad \forall .
A positive matrix is a matrix in which all the elements are strictly gre ...
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 process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
es are always Metzler matrices, and that probability distributions are always non-negative.
A Metzler matrix has an
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
in the nonnegative
orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutua ...
because of the corresponding property for nonnegative matrices.
Relevant theorems
*
Perron–Frobenius theorem
See also
*
Nonnegative matrices
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
*
Delay differential equation
In mathematics, delay differential equations (DDEs) 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.
DDEs are also called tim ...
*
M-matrix In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-p ...
*
P-matrix In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0.
Spectra of -matri ...
*
Z-matrix
*
Hurwitz matrix In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion
Namely, given a ...
*
Stochastic matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
*
Positive systems Positive systemsT. Kaczorek. Positive 1D and 2D Systems. Springer-
Verlag, 2002 constitute a class of systems that has the important property that its state variables are never negative, given a positive initial state. These systems appear frequentl ...
Bibliography
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Matrices
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