Editing Eigenfunction (section)

===Link to eigenvalues and eigenvectors of matrices===
Eigenfunctions can be expressed as column vectors and linear operators can be expressed as matrices, although they may have infinite dimensions. As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.

Define the [[inner product]] in the function space on which ''D'' is defined as
<math display="block"> \langle f,g \rangle = \int_{\Omega} \ f^*(t)g(t) dt,</math>
integrated over some range of interest for ''t'' called Ω. The ''*'' denotes the [[complex conjugate]].

Suppose the function space has an [[orthonormal basis]] given by the set of functions {''u''<sub>1</sub>(''t''), ''u''<sub>2</sub>(''t''), …, ''u''<sub>''n''</sub>(''t'')}, where ''n'' may be infinite. For the orthonormal basis,
<math display="block"> \langle u_i,u_j \rangle = \int_{\Omega} \ u_i^*(t)u_j(t) dt = \delta_{ij} =
\begin{cases}
  1 & i=j \\
  0 & i \ne j
\end{cases},</math>
where ''δ''<sub>''ij''</sub> is the [[Kronecker delta]] and can be thought of as the elements of the [[identity matrix]].

Functions can be written as a linear combination of the basis functions,
<math display="block">f(t) = \sum_{j=1}^n b_j u_j(t),</math>
for example through a [[Fourier series|Fourier expansion]] of ''f''(''t''). The coefficients ''b''<sub>''j''</sub> can be stacked into an ''n'' by 1 column vector {{nowrap|1=''b'' = [''b''<sub>1</sub> ''b''<sub>2</sub> … ''b''<sub>''n''</sub>]<sup>T</sup>}}. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite dimension.

Additionally, define a matrix representation of the linear operator ''D'' with elements
<math display="block"> A_{ij} = \langle u_i,Du_j \rangle = \int_{\Omega}\ u^*_i(t)Du_j(t) dt.</math>

We can write the function ''Df''(''t'') either as a linear combination of the basis functions or as ''D'' acting upon the expansion of ''f''(''t''),
<math display="block">Df(t) = \sum_{j=1}^n c_j u_j(t) = \sum_{j=1}^n b_j Du_j(t).</math>

Taking the inner product of each side of this equation with an arbitrary basis function ''u''<sub>''i''</sub>(''t''),
<math display="block">\begin{align}
  \sum_{j=1}^n c_j \int_{\Omega} \ u_i^*(t)u_j(t) dt &= \sum_{j=1}^n b_j \int_{\Omega} \ u_i^*(t)Du_j(t) dt, \\
  c_i &= \sum_{j=1}^n b_j A_{ij}.
\end{align}</math>

This is the matrix multiplication ''Ab'' = ''c'' written in summation notation and is a matrix equivalent of the operator ''D'' acting upon the function ''f''(''t'') expressed in the orthonormal basis. If ''f''(''t'') is an eigenfunction of ''D'' with eigenvalue λ, then ''Ab'' = ''λb''.