1
$\begingroup$

I'm newbie in Linear Algebra and have no knowledge in Differential equation. I'm studying Introduction to Linear algebra https://www.youtube.com/watch?v=13r9QY6cmjc in the video (at 6:00) the professor says $$AS = S \Lambda $$

Where A is a square matrix that has n independent eigenvector and S is a matrix of independent eigenvectors of A

then later in the video (at 27:30) when he solve $$u_{k+1} = Au_{k}$$

he goes into $$u_{k} = A^ku_{0}$$

Where $$u_{0} = c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+...c_{n}x_{n} = Sc$$

Then $$u_{k} = A^kSc = S\Lambda^kc$$ isn't it? why he writes $ A^{100}u_{0} = \Lambda^{100}Sc$ it suppose to be $ A^{100}u_{0} = S\Lambda^{100}c$ isn't it?

Is he write the wrong equation? and the $ A^{100}u_{0} = S\Lambda^{100}c$ is the correct one?

I see a lot of youtube comment said he was wrong on this but his document still make the same mistake as you can see on this link on page 2 in the section "Difference equations $u_{k+1} = Au_{k}$" the last line of the section writes $u_{k} = A^ku_{0} = c_{1}λ_{1}^kx_{1} +c_{2}λ_{2}^kx_{2} +···+c_{n}λ_{n}^kx_{n} = Λ^kSc$

which I think is wrong. It should be:

$u_{k} = A^ku_{0} = c_{1}λ_{1}^kx_{1} +c_{2}λ_{2}^kx_{2} +···+c_{n}λ_{n}^kx_{n} = SΛ^kc$

Am I right?

$\endgroup$
1
$\begingroup$

You are quite right (and the video is wrong on this). The point is that acting with $A$ upon the eigenvectors produce the eigenvectors times eigenvalue. And this written in matricial form is precisely $$ A S = S \Lambda$$ as you say. Similarly, $A^{100} S = S \Lambda^{100}$ etc...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.