Spectral Decomposition of Linear Operator T

[MathIdiot]

1. Let T be the linear operator on R^n that has the given matrix A relative to the standard basis. Find the spectral decomposition of T.

A=

7, 3, 3, 2
0, 1, 2,-4
-8,-4,-5,0
2, 1, 2, 3


3. eigen values are 1 (mulitplicity 1), -1 (mult. 1), 3 (mult. 2). And associated eigen vectors:

(1,-2,0,0)
(1,-6,4,-1)
(1,0,-1,-1/2), (0,1,-1/2,-3/4), respectively.

So, T = P1 - P2 + 3P3 (P1, P2, P3 being projection matrices)

I really need some sort of algorithm with perhaps this as an example, because I will have to solve more like it. Thanks so much!

 

[gumi_kr]

Try typing 'finding eigenvalues' in google and you will find your answer there. It is very common thing and you shall find it in any advanced 'linear algebra' book. (ctrl+f + eigenvalue)

The general idea of 'spectral decomposition' is to find eigenvalues and vectors associated with it (choose any of them), change base to eigenvectors and have a diagonal matrix.