Linear Algebra fun times.
Described with Latex below.

Question

Linear Algebra fun times. 
Described with Latex below.

sh3lsh · Answer

If one has a diagonalizable matrix A, one can reduce into $$S^{-1}AS=D$$
So, if we wanted to get A^5, would we just do this: 
$$S D^{5} S^{-1}$$

sh3lsh · Answer

Also, if we have a matrix A, when can we tell if it is diagonalizable or not?  (just point me to a link if adequate)

empty · Answer

I think to be diagonalizable you just have to have a nonzero determinant. I could be wrong though.

empty · Answer

I don't think there's a difference between having a complete set of linearly independent eigenvectors and having a nonzero determinant, they're essentially the same condition I believe.

sh3lsh · Answer

Eh? the way you calculate eigenvectors is to do to $$\det (A- \lambda I_{n})$$

empty · Answer

I think it must not only have a nonzero determinant but also have distinct eigenvalues in order to be diagonalizable. I'm sorta shaky on that, but I checked and found this: http://s-mat-pcs.oulu.fi/~mpa/matreng/eem4_2-3.htm Ok, it's a start

sh3lsh · Answer

(add an = 0 to that)

sh3lsh · Answer

Nonetheless was my way of calculating A^5 correct?

empty · Answer

For 2x2 matrices, the characteristic equation will be:
$$0=\lambda^2 - tr(A) \lambda + det(A)$$

So yes you definitely will use the determinant to calculate eigenvalues, although that's not really what I was saying to begin with anyways, I was just using it as a test to see if it had n eigenvalues.

@sh3lsh Yes that way is correct! This is how we can even calculate things like noninteger powers of matrices or weird things like $e^A$ with the Taylor series representation.

sh3lsh · Answer

Hmm. That person left and deleted their response! 
But thanks @Empty! I was just checking. You're awesome.

empty · Answer

Yeah weird I don't know why #_# I thought they had some good responses

empty · Answer

Diagonalization is also useful for transforming things like ellipses into circles, because these $S$ matrices will transform the vectors multiplying the quadratic form so that you don't have any cross terms, only the diagonal squared terms. Diagonalization is just an awesome thing to have really.

sh3lsh · Answer

Interesting. 
My professor doesn't tell us things like this! I appreciate this!

empty · Answer

Yeah if you wanna learn anything about linear algebra just ask I'd love to share and introduce some fun stuff