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Find the real and complex canoncial forms

  1. Dec 5, 2007 #1
    1. The problem statement, all variables and given/known data

    hey there. i have 3 equations in quadratic form:

    q1 [x] = [itex]x^2 + 2xy + 4yz + z^2[/itex]

    q2 [x] = [itex]2xy + 4yz - 2xz[/itex]

    q3 [x] = [itex](x + y + z)^2[/itex]

    2. What i need to find
    i have to find the real and complex canoncial forms:

    3. The attempt at a solution
    > i know i have to first turn the quadratic equations into matrix form.
    > Once ive done that, i get the real canonical form by converting all the positive terms into 1’s and the negative terms into -1’s.
    >The complex canonical form is then obtained by changing the minus 1’s to plus 1’s

    my attempt:

    so far i've managed to turn the first equation into a matrix:

    [1 1 0]
    [1 0 2]
    [0 2 1]

    i know i have to now diagonalize this matrix using "DOUBLE OPERATIONS" , consisting of a column operation followed by the "corresponding" row operation.
    but the problem is i just cant seem to finish this matrix off. does anyone please have the steps to diagonalize this using double operations?

    and also the steps to diagonalise the other 2 matrices? thanks a lot for the help :)
    Last edited: Dec 5, 2007
  2. jcsd
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