Eigenvectors and eigenvalues

[Andrew Dinn]

Andrew Dinn writes:

> Bonnie Surfus writes:

> > I am not yet sure what an eigenfunction is

Of course it all came flooding back as soon as I posted (I did this 16
years ago in school maths). An eigenfunction is an `annihilator
function' for a linear mapping constructed using the eigenvalues.
i.e. if mapping M has eigenvalues l1, ..., ln and associated
eigenvectors e1, ..., en then the eigenvectors satisfy the equations

    M ei = l1 * ei  for each i = 1, ..., n

or equivalently

    M ei - li * ei = 0

or factoring the multiplication

    (M - li) ei = 0

So, for any given eignevector ei the mapping (M - li) `annihilates'
the vector ei. If you mulitply (i.e. apply in sequence) the mappings

    (M - l1)(M - l2)...(M - ln)

then this product `annihilates' all the eigenvectors i.e. it maps all
eignevectors to zero. Given suitable conditions it actually maps any
vector to zero - in other words the product above is equivalent to a
null or zero mapping.

The eigenfunction is the algebraic polynomial

    (x - l1)(x -l2) ... (x -ln)

which expands out to something of the form

    x^n + a1 * x ^ n-1 + ... + an-1 * x + an

Substitute the mapping M for x and you get the zero mapping.
Substitute an eigenvalue for x and you get the value 0. So each
eigenvalue is a root of the eigenfunction (equivalently, each
eigenvalue is a zero of the annihilator functions).

n.b. sometimes the eigenvalue associated with an eigenvector is 0. In
such a case the `dud eigenvalue' does not contribute to the
annihilator function since it would merely add a redundant power of x
to each term.

> P.S. In German `eigen' means `self' or `own' (it functions as an
> adjective).

Someone mentioned that `eigen' was usually `translated' as
`characteristic'. Actually, the `eigenxxx' terms do not need
translating as they are not German in origin. Germans (so a colleague
assures me) find it very funny when they see this prefix in maths
classes. `Characteristic xxx' is a recent rewording.


Andrew Dinn
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there is no map / and a compass / wouldn't help at all