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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 25PM at BN3, Exam Forum

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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 $Av_{i}=\lambda _{i}v_{i}$ 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:
 $D=P^{1}AP$ and $A=PDP^{1}$.
 For any positive integer $k$ we have $A^{k}=PD^{k}P^{1}$ and $D^{k}={\begin{pmatrix}\lambda _{1}^{k}&&0\\&\ddots &\\0&&\lambda _{n}^{k}\end{pmatrix}}$.
 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.
 Assuming P is invertible, a proof of 1.
 Proof of 2.
 Example  the "reproduction of rabbits" matrix $A={\begin{pmatrix}0&1\\1&1\end{pmatrix}}$ (see the mathematica session below).
 Discussion of 3.
 The relationship with linear transformations and changes of basis.
 $v_{1}$ thorough $v_{n}$ form a basis and $P$ is invertible.