Editing Diagonal matrix (section)

== Operator theory ==
In [[operator theory]], particularly the study of [[PDEs]], operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a [[separable partial differential equation]]. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an [[integral transform]]—which changes the basis to an [[eigenbasis]] of [[eigenfunction]]s: which makes the equation separable. An important example of this is the [[Fourier transform]], which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the [[heat equation]].

Especially easy are [[multiplication operator]]s, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix.