Question

Hi there! Is there anyone who could help me understand Jordan normal/canonical form?

Answer 1

I found a fairly friendly article about it. I am not really sure, I've never really heard of this before.

http://math.berkeley.edu/~peyam/Math110Sp13/Handouts/Jordan%20Canonical%20Form.pdf

This might be a nice video to help as well:

https://www.youtube.com/watch?v=MTVqx1wh3Hs

Answer 2

Thanks. I've seen that video already, but unfortunately dr. bob never explains why it works... he just tells you it does work.

Answer 3

Well I'm just sort of wondering... What's the point of it? Why would you ever need something in Jordan Canonical form in the first place?

Answer 4

@Kainui repeated eigen values.

Answer 5

I need it for non diagonalizable matrices.

Answer 6

I believe the reasoning is somewhat similar to how come for partial fraction decomposition, you need to put \(Ax+B\) if your terms have more multiplicity. The exact reason I'm unsure though.

Answer 7

or at least, that's what I've been told.

Answer 8

I think the reason why it works would require quite a bit of time to explain. I've never seen it explained either.

Answer 9

So does it allow you to diagonalize non diagonalizable matrices? Fun.

Answer 10

As far as I know, yes.

Answer 11

Jordan Cannonical form doesn't actually diagonalize the matrix though. I mean the middle sparse matrix technically isn't a diagonal matrix so I'm not so sure how easy it is to raise it to a power. I think it's main purpose is for systems of first order linear differential equations.

Answer 12

ohw... Then I think it's not what I need. I'm trying to "interpolate" matrices by raising them to a power of any real number.

Answer 13

Well, Jordan canonical form might lead you to something like \[
\begin{bmatrix}
4&1 \\
0&4
\end{bmatrix}
\]How easy is it to raise this to a power?

Answer 14

11 ^ n
01

for example

would be

1n
01

Answer 15

well since your example isn't diagonalizable, I wouldn't know the answer.

Answer 16

Though mine isn't diagonalizable either :P