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Find the necessary and sufficient conditions on the real numbers a,b,c

  1. Feb 24, 2012 #1
    Find the necessary and sufficient conditions on the real numbers a,b,c for the matrix:
    \begin{bmatrix} 1 & a & b\\ 0 & 1 & c \\ 0 & 0 & 2 \end{bmatrix} to be diagonalizable.

    Attempt: Now for this one I also solved for the eigenvlues which were: λ1 = 1, λ2 = 1, λ3 = 2

    So the problematic eigenvalues will be the one of multiplicity 2, i.e λ = 1.

    So this means I'd have to obtain two linearly independent eigenvectors for λ = 1.

    I tried solving and got to this matrix: \begin{bmatrix} 0 & a & b \\ 0&0&c \\ 0&0 & 1 \end{bmatrix}

    But I won't be able to find two linearly independent eigenvectors from setting any of the variables equal to anything...I don't think. What's the next step?[/QUOTE]
     
  2. jcsd
  3. Feb 24, 2012 #2
    Re: Diagonalizability

    Take your matrix down to reduce row echelon form.

    From that point, you should be able to figure what one of the variables should be.
     
  4. Feb 24, 2012 #3
    Re: Diagonalizability

    Do you mean just reduce the variables: a , b, c? i.e: divide out a by 1/a, etc and row reduce those?
     
  5. Feb 24, 2012 #4
    Re: Diagonalizability


    Start with the 2nd and 3rd row first. Reduces those and then see what you can do to the top.
     
  6. Feb 24, 2012 #5
    Re: Diagonalizability

    \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} a row of zeros which means one free variable, or would this mean that a, c = 0 and b could be any value?
     
  7. Feb 25, 2012 #6
    Re: Diagonalizability

    Instead of dividing out the 1st row by a, leave it as a. You will still have a = 0.

    Why is c = 0? the second row means x_3 = 0.
     
  8. Feb 25, 2012 #7
    Re: Diagonalizability

    So then a = 0, x2 is a free variable, and x3 = 0. Now this leaves me with still only one free variable......actually......x1 is a free variable as well isn't it?......ahhhhhh, if that's the case I see what your getting at. But I didn't really state any conditions, all I did was solve a reduced matrix.

    Thanks
     
  9. Feb 25, 2012 #8
    Re: Diagonalizability

    What you know now is a must be 0. The question is can b and c be anything?
     
  10. Feb 25, 2012 #9
    Re: Diagonalizability

    Well if the same process would have to be applied to find any matrix that is diagonalizable, it must mean that b, c can be any real number because they will be eliminated through row reduction.
     
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