Easy linear algebra question

  1. given this matrix
    [tex] x_{1} + 2 x_{2} - 2x_{3} =7 [/tex]
    [tex] x_{1} + x_{2} + x_{3} =2 [/tex]
    [tex] 2x_{1} + 2x_{2} + x_{3} =5 [/tex]

    Show taht [itex] \rho(T_{g}) = 2 [/itex] where rho represenets the spectral radius for this matrix
    Tg represents the matrix formed from teh Gauss Seidel method

    i found Tg to be like this
    [tex] \left(\begin{array}{c|ccc}0&-2&-2&7\\-1&0&-1&2\\-2&-2&0&5\end{array}\right) [/tex]

    the Matrix Tg in question is
    [tex] \left(\begin{array}{ccc}0&-2&-2\\-1&0&-1\\-2&-2&0\end{array}\right) [/tex]


    spectral radius is the maximum of the eigenvalues. But for this matrix the eigenvalues i obtained were all zero. (Am i wrong here, do you wnat me to show the working?)
    So how can the spectral radius be 2??

    Please help! Your help is greatly appreciated!
     
  2. jcsd
  3. CarlB

    CarlB 1,204
    Science Advisor
    Homework Helper

    Your calculation of the eigenvalues is in error (at least). For example, (1,0,-1) has eigenvalue of 2.

    Carl
     
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