Continuous matrix = differential operator?

[nonequilibrium]

Hello,

Sorry if the question sounds silly, but can a continuous matrix be seen as a differential operator?

First of all, let me state that I have no idea what a continuous matrix would formally mean, but I would suppose there is such an abstract notion, somewhere?

Secondly, let me tell you where I'm coming from: I'm reading about dynamic processes and it seems they can sometimes be described by the notion of a "generator". For dynamic processes with a finite state space, the generator is a (normal) matrix (e.g. discrete space Markov processes). For dynamic processes with a continuous state space (e.g. Hamiltonian dynamics), the generator is a differential operator. My above question follows naturally from this observation.

 

[chiro]

Hey mr. vodka.

Differential operators are linear operators and thus they have the kind of form that a normal linear operator does.

http://en.wikipedia.org/wiki/Differential_operator

The thing however is how the operator is constructed and the contents of the operator. Are you familiar with differential geometry and tensor calculus?

Also there is an area of mathematics known as operator algebras which concerns the question of how to define the function of an operator. Think of for example how you deal the situation of calculating O² where O is a linear operator with the right properties.

What might be helpful is if you go back to the multivariable calculus definitions for a derivative in R^n as a matrix and then consider the construction of that matrix in terms of the fact that the matrix itself is a linear operator.

Also, take a look at operator algebras and how you can use them to solve differential equations where the state-space is actually a square matrix instead of a single number (so instead of 1x1 you have nxn matrix).