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# Eigenvalues / eigenvectors for squared matrix watch

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1. Hey I've been trying to do this question:

Find the eigenvalues and corresponding eigenvectors of the matrice , where

I've tried going about this the normal way [ det (A-λI) = 0 ] then finding the eigenvalues and eigenvectors correspondingly, but it runs out too complicated with huge numbers like 2304 etc. Is there an easier way to go about this?
2. I think the [ det (A-λI) = 0 ] method comes out nice.

I get eigenvalues -5,-8,-6
3. (Original post by insparato)
I think the [ det (A-λI) = 0 ] method comes out nice.

I get eigenvalues -5,-8,-6
You can always use those special properties (i forget them) which simplify calculating the determinant. I.e. calculate eigenvalues and corresponding eigenvectors of the matrice B, and then there is a formula to find the eigenvalues and corresponding eigenvectors of the matrice B^2.
4. Please, people, for my sanity, the word is matrix.
5. (Original post by generalebriety)
Please, people, for my sanity, the word is matrix.
Haha
6. (Original post by coffeym)
Haha
Sorry getting mixed up with my plurals there
7. Instead of all of this spamming, did you get the eigenvalues out using the det method?
8. (Original post by insparato)
Instead of all of this spamming, did you get the eigenvalues out using the det method?
I don't think it comes out nicely. At least, not without a fight.

Edit: I got a characteristic equation of t^3 + 75t^2 - 5296t + 83088 = 0. Which, according to some online cubic solver, isn't very nice. So yeah, I made a mistake somewhere. What was your method, Aaron?
9. (Original post by generalebriety)
I don't think it comes out nicely. At least, not without a fight.
I don't see any problem with the standard method (although either insparato or I have a sign error for one of the roots).

Of course, that's just the eigenvalues (and I can't be bothered to do the eigenvectors) for the matrix andrewlee89 provided.

I'm not sure how that's supposed to be related to a matrix . I'm guessing the matrix supposed to be B and we find need to find eigenvalues for B^2. In which case if v is an eigenvector for B, just apply B twice to see what happens with B^2.

10. (Original post by insparato)

But I thought he wanted the eigenvalues of B^2, where the above was B...

(Original post by DFranklin)
In which case if v is an eigenvector for B, just apply B twice to see what happens with B^2.
Good point.
11. Oh, my mind skipped that part! I'll have a go now. Im assuming the matrix given is B ? although that hasnt been said.
12. (Original post by insparato)
Oh, my mind skipped that part! I'll have a go now. Im assuming the matrix given is B ? although that hasnt been said.
No, it's poorly worded, but I think that's what he meant.

DFranklin's idea is probably best.
13. Yeah ive just had ago and looking at what David says i think that would be a better approach. By heck i need to do some matrix algebra.
14. (Original post by generalebriety)
No, it's poorly worded, but I think that's what he meant.

DFranklin's idea is probably best.
Oh my goodness so sorry! Yeah the matrix given was B=

And they want the corresponding eigenvalues and eigenvectors for . Sorry for the confusion. Any luck with the numbers sprawling into 4 digits ?
15. (Original post by andrewlee89)
Oh my goodness so sorry! Yeah the matrix given was B=

And they want the corresponding eigenvalues and eigenvectors for . Sorry for the confusion. Any luck with the numbers sprawling into 4 digits ?
Please tell me you're not actually working out B^2 and its characteristic polynomial...
16. (Original post by SsEe)
Please tell me you're not actually working out B^2 and its characteristic polynomial...
What other choice is there? I've been at this for the past two days (okay not yesterday, I took a rest) The problem is there's no answer in my book to this question sigh
17. (Original post by andrewlee89)
What other choice is there? I've been at this for the past two days (okay not yesterday, I took a rest) The problem is there's no answer in my book to this question sigh
Suppose Bv = kv (i.e. v is an eigenvector for B with eigenvalue k).

Can you work out what is? What does this tell you about the eigenvalues and eigenvectors of ?
18. Think about what an eigenvector is? There's probably some blurb before you come onto the subject explaining what's special about an eigenvector in a non-mathematical sense.
19. (Original post by DFranklin)
Suppose Bv = kv (i.e. v is an eigenvector for B with eigenvalue k).

Can you work out what is? What does this tell you about the eigenvalues and eigenvectors of ?
Hi, I'm new here, hello.

Sorry to resurrect an old thread, but would it be correct to say that ? I can't see how to prove this, my matrix algebra is really lacking. Could somebody show me?

Thanks.

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