06240/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.