## thdrbird Group Title Let A = [-3,2,6 0,-1,6 0,0,-3] Find an invertible matrix P and a diagonal matrix D such that D = P^{-1}AP. find P and D I worked out the polynomial -7x^3-7x^2-15x-9 to give x= -1,-3,-3 My answer for P= [1,0,0 1,-3,-3 0,1,1] and D= -1,0,0 0,-3,0 0,0,-3 something's wrong! one year ago one year ago

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1. electrokid Group Title

what type of decomposition are you doing here ?

2. Hoa Group Title

I think you get something wrong at P, since eigenvalue -3 has 2 dimensions and they are not yours! recheck

3. electrokid Group Title

you should get D=dia(-3,-1,-3)

4. Hoa Group Title

yes, surely, because they are eigenvalues of A, what I need is checking the order this guy arrange in P to have corresponding of numbers, like (-3,-1,-3) or (-1,-3,-3)

5. thdrbird Group Title

ok I will try!

6. Hoa Group Title

We cannot guess, right, kid? if we mess the order up, although we know it is the Diagonal matrix of A but the P is in different order of eigenvectors, We will be crazy when checking what wrong is, I had that experience. It made me crazy

7. electrokid Group Title

tr(A)=-3-1-3=-7 A11+A22+A33=3+9+3=15 |A|=-3(3)=-9 so, the characteristic polynomial would be$\Delta(t)=t^3+7t^2+15t+9$

8. electrokid Group Title

well, as long as the eigen vector columns are arranged in proper order, you should be good.

9. thdrbird Group Title

still no luck...

10. thdrbird Group Title

no

11. electrokid Group Title

eigen vectors=? for t=-1: $-2x+2y+6z=0\\6z=0\\-3z=0\implies v_1=(1,1,0)^T$

12. thdrbird Group Title

yup

13. thdrbird Group Title

and v2 and v3 I put at (0,-3,1)

14. electrokid Group Title

for t= -3: $2y+6z=0\\ 2y+6z=0\\ v_{2,3}=(\alpha,-3\beta,\beta)^T$

15. electrokid Group Title

@thdrbird you cannot have them two identical

16. thdrbird Group Title

oh,ok..

17. electrokid Group Title

you can use v2 when a=0,b=1 v3 when a=1,b=1

18. electrokid Group Title

or v3 when a=1,b=0 ANY combination of alpha and beta that do not give the same vectors..

19. thdrbird Group Title

can you give me an example?

20. thdrbird Group Title

got it!

21. thdrbird Group Title

I forgot to set alpha!

22. thdrbird Group Title

thanks @electrokid, great help!

23. thdrbird Group Title

@Hoa as well! Thanks!

24. Hoa Group Title

I did nothing guy!!!