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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} matrix (with entries in some field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1,\ldots,v_n} be corresponding eigenvectors, so that $Av_{i}=\lambda _{i}v_{i}$ for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\leq i\leq n} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} be the diagonal matrix that has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1} through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_n} on its main diagonal (in order) and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} be the matrix whose columns are these eigenvectors: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P:=(v_1|v_2|\cdots|v_n)} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is invertible and the following equalities hold:

1. $D=P^{-1}AP$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=PDP^{-1}} .
2. For any positive integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^k=PD^kP^{-1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D^k=\begin{pmatrix}\lambda_1^k&&0\\&\ddots&\\0&&\lambda_n^k\end{pmatrix}} .
3. Likewise if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F={\mathbb R}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(B):=\sum_{k=0}^\infty\frac{B^k}{k!}} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(A)=P\exp(D)P^{-1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $A={\begin{pmatrix}0&1\\1&1\end{pmatrix}}$ (see the mathematica session below).
4. Discussion of 3.
5. The relationship with linear transformations and changes of basis.
6. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1} thorough Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_n} form a basis and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is invertible.