# Find the real and complex canoncial forms

1. Dec 5, 2007

### smoothman

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

hey there. i have 3 equations in quadratic form:

q1 [x] = $x^2 + 2xy + 4yz + z^2$
[y]
[z]

q2 [x] = $2xy + 4yz - 2xz$
[y]
[z]

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

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:
i.e.

[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