# 06-240/Classnotes For Tuesday December 5

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

Theorem. Let ${\displaystyle A}$ be an ${\displaystyle n\times n}$ matrix (with entries in some field ${\displaystyle F}$) and let ${\displaystyle \chi _{A}(\lambda ):=\det(A-\lambda I)}$ be the characteristic polynomial of ${\displaystyle A}$. Assume ${\displaystyle \chi _{A}}$ has ${\displaystyle n}$ distinct roots ${\displaystyle \lambda _{1}\ldots \lambda _{n}}$, that is, ${\displaystyle A}$ has ${\displaystyle n}$ distinct eigenvalues ${\displaystyle \lambda _{1}\ldots \lambda _{n}}$, and let ${\displaystyle v_{1},\ldots ,v_{n}}$ be corresponding eigenvectors, so that ${\displaystyle Av_{i}=\lambda _{i}v_{i}}$ for all ${\displaystyle 1\leq i\leq n}$. Let ${\displaystyle D}$ be the diagonal matrix that has ${\displaystyle \lambda _{1}}$ through ${\displaystyle \lambda _{n}}$ on its main diagonal (in order) and let ${\displaystyle P}$ be the matrix whose columns are these eigenvectors: ${\displaystyle P:=(v_{1}|v_{2}|\cdots |v_{n})}$. Then ${\displaystyle P}$ is invertible and the following equalities hold:

1. ${\displaystyle D=P^{-1}AP}$ and ${\displaystyle A=PDP^{-1}}$.
2. For any positive integer ${\displaystyle k}$ we have ${\displaystyle A^{k}=PD^{k}P^{-1}}$ and ${\displaystyle D^{k}={\begin{pmatrix}\lambda _{1}^{k}&&0\\&\ddots &\\0&&\lambda _{n}^{k}\end{pmatrix}}}$.
3. Likewise if ${\displaystyle F={\mathbb {R} }}$ and ${\displaystyle \exp(B):=\sum _{k=0}^{\infty }{\frac {B^{k}}{k!}}}$ then ${\displaystyle \exp(A)=P\exp(D)P^{-1}}$ and ${\displaystyle \exp(D)={\begin{pmatrix}e^{\lambda _{1}}&&0\\&\ddots &\\0&&e^{\lambda _{n}}\end{pmatrix}}}$.

Order of the proceedings.

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