power of a matrix
https://drive.google.com/file/d/0B7ss_Y8M_VEgdWt2czBpT1h0YkU/view?usp=drivesdk

Question

power of a matrix
https://drive.google.com/file/d/0B7ss_Y8M_VEgdWt2czBpT1h0YkU/view?usp=drivesdk

ganeshie8 · Answer

I think the last line in above screenshot is wrong.
It should be $S\Lambda^{100}c$

kainui · Answer

Kinda hard for me to tell what's going on, but looks like they have:
$$A = S \Lambda S^{-1}$$
then they're squaring it I assume:

$$A^2 = S\Lambda S^{-1} S \Lambda S^{-1} = S \Lambda^2 S^{-1}$$
kinda like that and cause $\Lambda$ is a diagonal matrix you end up with the entries raised to that power which is simple, so maybe this is like a proof of that? (just my guess)

kainui · Answer

Or wait why am I guessing haha, can you send the link to the OCW vid, looks like this is at 32:51 I'll just check that out real fast.

ganeshie8 · Answer

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-22-diagonalization-and-powers-of-a/

ganeshie8 · Answer

I don't get how the diagonal matrix is placed to the left of the eigemvector matrix in the last line in my attached pic

ganeshie8 · Answer

My question is about how the last line was derived from the line before it..

ganeshie8 · Answer

In your formula for power of A also, S is the left most matrix. But in the attached pic, diagonal matrix is placed left of S.

ganeshie8 · Answer

I think strange made a typo..

kainui · Answer

Haha Strang* made a typo XD

kainui · Answer

Yeah I see what you're saying that should really be $S^{-1}$ it looks like, I think you're right I'm looking through it

ganeshie8 · Answer

thank you
I just want to get convinced that I'm not doing the multiplication wrong...

kainui · Answer

When I think of eigenvectors I imagine this equation:

$$A x_i = x_i \lambda_i$$
which then if I want to think of all the eigenvalues and eigenvectors (column vectors) put side by side in matrices we get:

$$A S = S \Lambda$$

so when he writes:

$$u_0 = c_1x_1 + \cdots + c_n x_n = Sc$$

That much is correct since c is a column vector weighing all these column eigenvectors of S.

But then after that he multiplies it by $\Lambda$ which seems like it should put a weight on each one, however it's wrong. What he should have done is:

$$S\Lambda^{100} c$$

ganeshie8 · Answer

Exactly! Thanks for confirming, I can sleep happily tonight :)

kainui · Answer

Awesome, yeah eigenvectors are just so cool, diagonalization is just so insanely powerful it should be as well respected as calculus I think sometimes lol.