# 06-240/Classnotes For Tuesday December 5

Our remaining goal for this semester is to study the following theorem:

Theorem. Let A be an $n\times n$ 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 $\lambda_1\ldots\lambda_n$, that is, A has n distinct eigenvalues $\lambda_1\ldots\lambda_n$, and let $v_1,\ldots,v_n$ be corresponding eigenvectors, so that Avi = λivi for all $1\leq i\leq n$. 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: $P:=(v_1|v_2|\cdots|v_n)$. Then P is invertible and the following equalities hold:

1. D = P − 1AP and A = PDP − 1.
2. For any positive integer k we have Ak = PDkP − 1 and $D^k=\begin{pmatrix}\lambda_1^k&&0\\&\ddots&\\0&&\lambda_n^k\end{pmatrix}$.
3. Likewise if $F={\mathbb R}$ and $\exp(B):=\sum_{k=0}^\infty\frac{B^k}{k!}$ then exp(A) = Pexp(D)P − 1 and $\exp(D)=\begin{pmatrix}e^{\lambda_1}&&0\\&\ddots&\\0&&e^{\lambda_n}\end{pmatrix}$.

Order of the proceedings.

1. Assuming P is invertible, a proof of 1.
2. Proof of 2.
3. Example - the "reproduction of rabbits" matrix $A=\begin{pmatrix}0&1\\1&1\end{pmatrix}$ (see the mathematica session below).
4. Discussion of 3.
5. The relationship with linear transformations and changes of basis.
6. v1 thorough vn form a basis and P is invertible.