06-240/Classnotes For Tuesday December 5
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.