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Spectral Decomposition of Linear Operator T

  1. Dec 15, 2008 #1
    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. jcsd
  3. Dec 26, 2008 #2
    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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