06-240/Classnotes For Tuesday December 5

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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 $\chi_A(\lambda):=\det(A-\lambda I)$ be the characteristic polynomial of $A$. Assume $\chi_A$ 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 $Av_i=\lambda_iv_i$ for all $1\leq i\leq n$. Let $D$ be the diagonal matrix that has $\lambda_1$ through $\lambda_n$ 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. $D=P^{-1}AP$ and $A=PDP^{-1}$.
2. For any positive integer $k$ we have $A^k=PD^kP^{-1}$ 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 $\exp(A)=P\exp(D)P^{-1}$ 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. $v_1$ thorough $v_n$ form a basis and $P$ is invertible.