Linear Transforms, Eigen Functions, Hermitian Operators
Eigen Functions
Let t be a transform from a function space into itself,
let f be a function, and let c be a scalar,
such that t(f) = c×f.
Since t operates on a vector space, f is an eigen vector and c is an eigen value.
Yet f is also a function,
so to keep things clear,
f is sometimes called an eigen function.
Most of the theorems regarding eigen vectors apply to eigen functions,
since they are, after all, vectors in a vector space.
However, the dimension of the space is infinite,
so a theorem that relies on n×n matrices may not carry over.
One of the concepts that remains applicable is the description of an eigen space.
For any given eigen value c,
the set of eigen functions that are scaled by c forms a vector space,
also called an eigen space.
The dimension of this space could be finite, or infinite.
What are the eigen functions under the differentiation operator?
Solve the differential equation y′ = cy,
giving kEcx.
Here c is the eigen value,
and the eigen space for c has one dimension, as indicated by the secondary constant k.
Dot Product
The dot product, or inner product, of two vectors
is the sum of the pairwise products of corresponding components.
If entries are complex, the second vector is conjugated before multiplication.
Thus u.v is the conjugate of v.u.
The dot product generalizes to continuous functions in a natural manner.
The dot product of f and g is the integral of f times g,
or the integral of f times the conjugate of g if g is complex.
This is well defined over a closed interval such as [0,1].
It is also well defined from -∞ to +∞ for certain classes of functions with tails.
Most of these generalizations were discussed when we
introduced the
dot product,
and you should probably review that section before proceeding.
Hermitian Operators
Once a valid dot product has been established,
an operator t is hermitian if t(f).g = f.t(g) for every function f and g.
The operator is skew hermitian if t(f).g = -f.t(g).
If the functions are real (not complex),
these operators are called symmetric and skew symmetric respectively.
In finite dimensional space, a transformation is symmetric iff its matrix is symmetric,
and a transformation is hermitian iff its matrix is hermitian.
(A hermitian matrix is equal to the conjugate of its transpose.)
But we're not in finite dimensional space any more, and there is no matrix,
so we use the dot product definition of hermitian and skew hermitian.
You can view some important theorems on hermitian operators,
and some helpful examples, by going here.
Orthogonal Eigen Functions
If t is a hermitian transform, all its eigen values are real,
and eigen functions with distinct eigen values are orthogonal,
i.e. their dot product is 0.
If t is a skew hermitian transform, all its eigen values are imaginary,
and eigen functions with distinct eigen values are orthogonal.
Review the proof here.
As you can see, there aren't any new theorems on this page.
I just wanted to review some topics in linear algebra that generalize to
linear transforms over an infinite dimensional space.