06-240/Classnotes For Tuesday December 5
Our remaining goal for this semester is to study the following theorem:
Theorem. Let A be an matrix (with entries in some field F) and let χA(λ): = det(A − λI) be the characteristic polynomial of A. Assume χA has n distinct roots , that is, A has n distinct eigenvalues , and let be corresponding eigenvectors, so that Avi = λivi for all . Let D be the diagonal matrix that has λ1 through λn on its main diagonal (in order) and let P be the matrix whose columns are these eigenvectors: . Then P is invertible and the following equalities hold:
- D = P − 1AP and A = PDP − 1.
- For any positive integer k we have Ak = PDkP − 1 and .
- Likewise if and then exp(A) = Pexp(D)P − 1 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.
- v1 thorough vn form a basis and P is invertible.