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Linear algebra diagonalization

  1. Apr 27, 2009 #1
    linear algebra diagonalization :(

    1. The problem statement, all variables and given/known data

    determine whether the given matrix is diagonalizable. where possible, find a matrix S such that
    S-1AS=Diag(λ12,...,λn)

    2. Relevant equations



    3. The attempt at a solution

    I was able to find the eigenvalues, which are λ=1,-4,4. This is given in the back of the book as well, which matched up evenly.

    Now, I'm having trouble finding the matrix S. I know I need to find the eigenvectors, and place them as columns in an n x n matrix.

    When I plug 1 in, say, I come up with
    0 0 0
    0 2 7
    1 1 -4

    And for 4 I came up with
    -3 0 0
    0 -1 7
    1 1 -7

    Neither of them seem to be coming out right.

    The answer in the back of the book is:

    15 0 0
    -7 7 1 = S
    2 1 -1

    Thanks, all.
     
  2. jcsd
  3. Apr 28, 2009 #2
    Re: linear algebra diagonalization :(

    let's see your rref. what is the nullspace for each of these matrices?
     
  4. Apr 28, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: linear algebra diagonalization :(

    With [itex]\lambda= 1[/itex] your matrix requires that 0x+ 0y+ 0z= 0, 0x+ 2y+ 7z= 0, and x+ y- 4z= 0. The first equation is always true, of course, the second equation gives z= -(7/2)y and putting that into the third equation x- (7/2)z- 4z= x- (15/2)z= 0 so x= (15/2)z. Taking z= 2, an eigenvector corresponding to [itex]\lambda= 1[/itex] is <15, -7, 2>. That is exactly what you say your textbook gives as the first column.

    With [itex]\lambda= 4[/itex] the corres0 7 ponding equations are -3x= 0, 0x- y+ 7z= 0, and x+ y- 7z= 0. The first equation obvioiusly gives x= 0. Putting x= 0, the second and third equations both reduce to y- 7z= 0 or y= 7z. Taking z= 1 y= 7 so an eigenvalue corresponding to [itex]\lambda= -4[/itex] is <0, 7, 1> , exactly what your textbook gives for the second column.

    With [itex]\lamba= 4[/itex] the matrix [itex]A- \lambda I[/itex] must be
    [tex]\begin{bmatrix}5 & 0 & 0 \\ 0 & 7 & 7 \\ 1 & 1 & 1\end{bmatrix}[/tex]
    which gives equations 5x+ 0y+ 0z= 0, 0x+ 7y+ 7z= 0, and x+ y+ z= 0. What eigevector do those give?
     
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