Question

Let v be an eigenvector of the matrix M, with corresponding eigenvalue λ. Show that v is
also an eigenvector of the matrix M^2 with corresponding eigenvalue λ^2.

Your kind assistance is always appreciated. Many Thanks...

Answer 1

I mean M

Answer 2

so what is M^2

Answer 3

Something happened to my equation

Answer 4

Let L = the diagonal matrix with the eigenvalues

Answer 5

I have no idea ;(
i saw it briefly then it disappeared. :-)

Answer 6

ok

Answer 7

v L v^-1 = M
v L v^-1 v L v^-1 = M M

Answer 8

Whaddya got?

Answer 9

is v the eigenvector and are you multiplying it with - landa v_1 ?

Answer 10

Yes
By definition v lambda v^-1 = M

Answer 11

okay, and M is a normal matrix correct?

Answer 12

normal? I think it has to be square.

Answer 13

sorry but what is v^-1 ?

Answer 14

yes i mean square 2x2 , 3x3 nxn

Answer 15

You get your eigenvectors and put em in a matrix and then invert it.

Answer 16

right?

Answer 17

Sure

Answer 18

That's the whole idea v L v^-1 = M

Answer 19

and is that all two or all three at a time?

Answer 20

So you can chain them , that's what's great.

Answer 21

yes

Answer 22

The v v^-1 all cancel.

Answer 23

right

Answer 24

So then we have
M^2 = v L^2 v^-1

Answer 25

okay

Answer 26

eigenvectors right there

Answer 27

squared eigenvalues in a diagonal matrix

Answer 28

I might have to read up on this a bit more...under what specific topic can I find this ? Is it under properties of eigenvectors/eigenvalues?

Answer 29

Sure. I have trouble remembering which lecture, but I learned this from Gil Strang's MIT ocw course in linear algebra. He's the best teacher I have ever seen.

Answer 30

was that the online course?

Answer 31

Yep

Answer 32

i will have a look, do u think that khanacademy covers this?

Answer 33

I'm pretty sure he does. Advantage of Khan is short and sweet. Disadvantage is he is fragmented and sometimes makes mistakes. Your pref.

Answer 34

I think it might fall under the definition - similar matrices, I must have a look... I love MIT though, and its great when you have loads of time...

Answer 35

I will get back to you tomorrow with chapter. Are you good with basic facts about Linear Algebra?

Answer 36

I'm getting better, but need to revise here and there....I'm finishing 1st year engineering so I will have to be good with basic facts ...hehehe

Answer 37

Yes. good luck.

Answer 38

I will watch out for you tomorrow. Will you msh me the chapter please. Much Appreciated...thank you . for your kind help

Answer 39

Tell you what, I will explain Linear Algebra
You can explain Fourier series and Laplace Transform. !!!

Answer 40

hehehe...At the end of year 2!!! Thats 12 months from now!