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Math Help - Find out eigenvector of this matrix

  1. #1
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    Find out eigenvector of this matrix

    Hi, I want to find out the eigenvector of this matrix, but I am not sure how, because there are some zeros in the columns:
    [-7/10 0 0
    7/10 -3/10 0
    0 3/10 -1/10]
    The formula came out to be:
    [0 0 0
    0 4/10 0
    0 3/10 6/10]
    I used the online calculator to found that the eigenvectors are
    (8, -14, 7) for -7/10
    (0, -2, 3) for -3/10
    (0, 0, 1) for -1/10
    Can anyone show me the arithmatic (just one is enough for an example)

    Thanks,
    Ted
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  2. #2
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    Tejas
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    normally, i find eigenvectors by finding the eigenvalues first. this involves finding det(A - xI) for your matrix A.

    in this particular case, the 0's actually make it easier to take the determinant, and det(A - xI) =

    -(7/10 + x)(3/10 + x)(1/10 + x). setting this equal to 0 gives the 3 eigenvalues: -7/10, -3/10 and -1/10.

    so let's see what we can come up with for the eigenvalue -7/10. we know that for an eigenvector v for the eigenvalue -7/10

    Av + (7/10)v = 0. in matrix form:

    [..0.. ..0.. .....0][v1].....[0]
    [7/10 4/10 ....0][v2]....[0]
    [..0.. 3/10 6/10][v3] = [0]

    since the first row is all 0, it doesn't tell us anything, the first entry of (A +(7/10)I)v will be 0 no matter what v is.

    but from the 2nd row we see that (7/10)v1 + (4/10)v2 = 0, so v1 = (-4/7)v2.

    and from the 3rd row, we have (3/10)v2 + (6/10)v3 = 0, so v3 = (-1/2)v2.

    we can thus pick any value we like for v2, and this will determine v1 and v3. to avoid fractions, let's pick v2 = -14.

    then v1 = (-4/7)v2 = (-4/7)(-14) = 8, and v3 = (-1/2)(-14) = 7, so v = (8,-14,7).

    ( (-8,14,-7) and (8/7,-2,1) would also be perfectly acceptable eigenvectors).

    the other two eigenvectors can be calculated in the same way.
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  3. #3
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    Thanks Deveno! You are awesome! I guess the picking your own v2 was the part that I am stuck for
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