# Spectral Decomposition of Linear Operator T

1. Dec 15, 2008

### 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!!

2. Dec 26, 2008

### 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.