Linear operators and matrices are very much related. Given two finite-dimensional vector spaces
U,V over a field
K, and
E={ei∣i∈{1,...,n}},
F={fi∣i∈{1,...,n}} bases of
U and
V respectively, for any linear
T[s]u[/s]→V, let
ci be
T(ei) expressed as a vector in terms of
F for each
i. Then the matrix
A with the
ci as its columns is the matrix of
T, and they behave in a very similar way: that is, if
u∈U, and
u∗ is
u expressed as a vector in terms of the
ei, then
T(u), expressed in terms of the
fi is exactly
Au∗. (Similarly, you can start with the matrix
A and, given the bases, get
T back, by simply taking the columns of
A as the images of the
ei).
Once you've got this, the eigenvalues and eigenvectors of
T are exactly the eigenvectors and eigenvalues of
A. Eigenfunctions are a special case of eigenvectors, where
U is a function space.
You can also define eigenvalues and eigenvectors of a linear operator directly, in the same way as you do with matrices:
With
T,U as above, with
V=U, the eigenvectors of
T are the non-zero vectors
u∈U such that
T(u)=λu for some
λ∈K, and the eigenvalues of
T are the
λ∈K such that there is some
u∈U such that
T(u)=λu.