06-240/Classnotes For Tuesday December 5
From Drorbn
|
Our remaining goal for this semester is to study the following theorem:
Theorem. Let be an
matrix (with entries in some field
) and let
be the characteristic polynomial of
. Assume
has
distinct roots
, that is,
has
distinct eigenvalues
, and let
be corresponding eigenvectors, so that
for all
. Let
be the diagonal matrix that has
through
on its main diagonal (in order) and let
be the matrix whose columns are these eigenvectors:
. Then
is invertible and the following equalities hold:
-
and
.
- For any positive integer
we have
and
.
- Likewise if
and
then
and
.
Order of the proceedings.
- Assuming P is invertible, a proof of 1.
- Proof of 2.
- Example - the "reproduction of rabbits" matrix
(see the mathematica session below).
- Discussion of 3.
- The relationship with linear transformations and changes of basis.
-
thorough
form a basis and
is invertible.