06-240/Classnotes For Tuesday December 5

06-240/Navigation Panel   # Week of... Notes and Links 1 Sep 11 About, Tue, HW1, Putnam, Thu 2 Sep 18 Tue, HW2, Thu 3 Sep 25 Tue, HW3, Photo, Thu 4 Oct 2 Tue, HW4, Thu 5 Oct 9 Tue, HW5, Thu 6 Oct 16 Why?, Iso, Tue, Thu 7 Oct 23 Term Test, Thu (double) 8 Oct 30 Tue, HW6, Thu 9 Nov 6 Tue, HW7, Thu 10 Nov 13 Tue, HW8, Thu 11 Nov 20 Tue, HW9, Thu 12 Nov 27 Tue, HW10, Thu 13 Dec 4 On the final, Tue, Thu F Dec 11 Final: Dec 13 2-5PM at BN3, Exam Forum Register of Good Deeds Add your name / see who's in! edit the panel

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 $Av_{i}=\lambda _{i}v_{i}$ 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 $A^{k}=PD^{k}P^{-1}$ and Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle D^k=\begin{pmatrix}\lambda_1^k}