Find the real and complex canoncial forms

[smoothman]

Homework Statement

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:

The Attempt at a Solution

> i know i have to first turn the quadratic equations into matrix form.
> Once I've 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 can't 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 :)

 

[mighty2000]

Thank you for sharing your equations and your attempt at finding the real and complex canonical forms. I would like to offer some guidance and suggestions to help you with your problem.

Firstly, you are on the right track by converting the quadratic equations into matrix form. This will make it easier to perform the necessary operations to find the canonical forms. However, I would like to point out that your first matrix is not in the correct form. The matrix form for a quadratic equation should have the coefficients of the x^2, xy, y^2, etc. terms as the entries of the matrix. So for your first equation, the matrix form should be:

[1 2 0]
[2 0 4]
[0 4 1]

I would suggest double checking the matrix forms for the other two equations as well.

Next, to find the real canonical form, you are correct in converting all the positive terms to 1's and negative terms to -1's. However, to do so, you will need to use the eigenvalues and eigenvectors of the matrix. The diagonal elements of the real canonical form will be the eigenvalues of the matrix, while the eigenvectors will form the transformation matrix that will convert the original matrix into the canonical form.

To find the complex canonical form, you will need to use the Cayley-Hamilton theorem, which states that a matrix satisfies its own characteristic equation. This will help you find the powers of the matrix that will give you the complex canonical form.

As for the specific steps to diagonalize the matrices using double operations, I would suggest consulting a linear algebra textbook or online resources for a detailed explanation. It involves finding the eigenvalues and eigenvectors of the matrix, and then using them to perform the necessary operations to convert the matrix into its canonical form.

I hope this helps guide you in the right direction. Good luck with your calculations!