Another spectral decomp question.

[hank]

Using PDP^-1 method.

Ok, I got this question wrong on a test.

Essentially, I had to find A = E1 + E2 where

A =
1 2 -2
0 5 -4
0 2 -1

satisfies

P^-1AP=
1 0 0
0 1 0
0 0 3

with P =
2 1 1
1 1 2
1 1 1

Ok, so..

E1 = PD1P^-1

E2 = PD2P^-1

Now, the correct answer shows:

D1 =
|1 0 0 |
|0 1 0|
|0 0 0|

D2 =
|0 0 0|
|0 0 0|
|0 0 1|


My question is why? Why couldn't it have been the other way around?

 

[Deleted member 4993]

Using PDP^-1 method.

Ok, I got this question wrong on a test.

Essentially, I had to find A = E1 + E2 where

A =
1 2 -2
0 5 -4
0 2 -1

satisfies

P^-1AP=
1 0 0
0 1 0
0 0 3

with P =
2 1 1
1 1 2
1 1 1

Ok, so..

E1 = PD1P^-1

E2 = PD2P^-1

Now, the correct answer shows:

D1 =
|1 0 0 |
|0 1 0|
|0 0 0|

D2 =
|0 0 0|
|0 0 0|
|0 0 1|


My question is why? Why couldn't it have been the other way around?

What exactly do you mean by "other way around"?

 

[hank]

Why is the diagonal for D1 1,1,0 and D2 0,0,1?
Why isn't it 0,0,1 for D1 and 1,1,0 for D2?