Re: Checking names of eigenvalue "multiplicities"

On Jul 29, 9:43 pm, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
Just doing a bit of work on linear algebra again. The book
by Axler which I'm using just defines the "multiplicity" of
an eigenvalue e as the dimension of its space of generalised
eigenvalues, which is equal to the exponent of z - e in the
characteristic polynomial q(z) (at least for a complex space
- perhaps algebraically closed in general, but Axler doesn't
treat the general case).

After looking in a couple of other books, I think that this
is usually called the "algebraic" multiplicity, while the
"geometric" multiplicity is the dimension of the null space
of T - eI, where T is the operator of which e is an eigen-
value. Is that right?

The solution of Exercise 23 in Chapter 8 seems to require me
to work with the exponent of z - e in the minimal polynomial
p(z) of T. None of my books seems to mention this number,
but a quick look at the Web suggests that it is called the
"index" of e. Is that right? If so, funny that it's not
called some other sort of "multiplicity", especially as an
exponent of a linear factor of a polynomial is called that
(or so I seem to recall).

(Hope this is clear - it's a bit late at night here.)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril

Correct, the geometric multiplicity is the dimension of
the subspace spanned by the eigenvectors of given eigen-
value (nullity of T - lambda I), while the algebraic
multiplicity is the dimension of the subspace spanned by
corresponding generalized eigenvectors (annihilated by
some power of T - lambda I, not necessarily the first).

regards, chip

.

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