Question

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!

Answer 1

what type of decomposition are you doing here ?

Answer 2

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

Answer 3

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

Answer 4

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)

Answer 5

ok I will try!

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

still no luck...

Answer 10

no

Answer 11

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

Answer 12

yup

Answer 13

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

Answer 14

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

Answer 15

@thdrbird you cannot have them two identical

Answer 16

oh,ok..

Answer 17

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

Answer 18

or v3 when a=1,b=0

ANY combination of alpha and beta
that do not give the same vectors..

Answer 19

can you give me an example?

Answer 20

got it!

Answer 21

I forgot to set alpha!

Answer 22

thanks @electrokid, great help!

Answer 23

@Hoa as well! Thanks!

Answer 24

I did nothing guy!!!