Spectral Decomposition of Linear Operator T

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The spectral decomposition of the linear operator T, represented by the matrix A, reveals eigenvalues of 1 (multiplicity 1), -1 (multiplicity 1), and 3 (multiplicity 2), with corresponding eigenvectors provided. The decomposition can be expressed as T = P1 - P2 + 3P3, where P1, P2, and P3 are projection matrices associated with the eigenvalues. To perform spectral decomposition, one must find the eigenvalues and their associated eigenvectors, then change the basis to the eigenvectors to form a diagonal matrix. For those seeking algorithms or methods to solve similar problems, resources on finding eigenvalues in advanced linear algebra texts or online searches are recommended. Understanding these concepts is crucial for effectively applying spectral decomposition in linear algebra.
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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!
 
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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.
 

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