Completing the square using a matrix in quadratic form

[PhizKid]

Homework Statement

Complete the square using the symmetric matrix that defines the given quadratic form: $x^2 - 4xy + 6xz + 2xt + 4y^2 + 2yz + 4yt + 5z^2 - 6zt - t^2$ and write this quadratic as the sum and difference of squares after completing the square using the matrix.

The Attempt at a Solution

So first I found the 4x4 matrix, using x as the first column, y as the 2nd, z as the 3rd, and t as the 4th. Also x is 1st order, y is 2nd order, z is 3rd, and t is 4th:

http://i2.minus.com/inT7VWSkOu3GH.png

If I want this in reduced row form, I have to switch some rows, name the 2nd and 3rd rows. What implications will this have on my work in completing the square?

Also, I couldn't find anything online about how to complete the square using matrices.

 

[HallsofIvy]

You don't want to reduce it. You want to find the eigenvalues and eigenvectors of the matrix and those are not "preserved" by row operations.

 

[PhizKid]

The assignment says to row eliminate into the row echelon form and use this row echelon form of the matrix to complete the square. Is there a method to complete the square using the row echelon form?

 

[Ray Vickson]

The assignment says to row eliminate into the row echelon form and use this row echelon form of the matrix to complete the square. Is there a method to complete the square using the row echelon form?

If I were doing the problem I would write
$$Q_4(x,y,z,t) = x^2 - 4xy + 6xz + 2xt + 4y^2 + 2yz + 4yt + 5z^2 - 6zt - t^2$$
as
$$Q_4(x,y,z,t) = u_1(x,y,z,t)^2 + Q_3(y,z,t)$$
by noting that
$$S_n = a_{11}x_1^2 + 2 a_{12} x_1 x_2 + \cdots + 2 a_{1n} x_1 x_n \\ = \left( \sqrt{a_{11}}x_1 + \frac{a_{12}}{\sqrt{a_{11}}} x_2 + \cdots + \frac{a_{1n}}{\sqrt{a_{11}}} x_n \right)^2 - S_{n-1},$$
where $S_{n-1}$ is a quadratic form not involving $x_1$. It has the form
$$S_{n-1} = (a_{12}^2 /a_{11}) x_2^2 + \cdots + (a_{1n}^2/a_{11})x_n^2 + 2 (a_{12}a_{13}/a_{11}) x_2 x_3 + \cdots + 2(a_{1,n-1}a_{1n}/a_{11})x_{n-1} x_n$$
So,
$$u_1 = x - 2 y + 3z + t,$$
and you can figure out what is $Q_3(y,z,t)$. Do the same type of operation on $Q_3$. If you never take the square root of a negative number throughout, you will reduce your quadratic form to a sum of squares. If you encounter the square root of a negative number at some point, you get a difference of (real) squares.

This method is attempting to write your 4x4 matrix $A$ in the form $A = U^T U,$ where $U$ is an upper-triangular matrix. I suppose you could regard $U^T$ as a matrix in row-echelon form, so I suppose that could be what the question is asking about.

 

[Honors LinAlg]

To do this by finding RREF, you want to use the form

E_n^T...E₂^TE₁^TDE₁E₂...E_n

Where D = diag(eigenvalues), which is found by solving E_n...E₂E₁AE₁^TE₂^T...E_n^T = A_RREF. In order to find D, take the column & row reductions one at a time - noting that you have to do AEE^T for every row manipulation E. (E is an elementary matrix s.t. det(E) = 1)

 

[Ray Vickson]

To do this by finding RREF, you want to use the form

E_n^T...E₂^TE₁^TDE₁E₂...E_n

Where D = diag(eigenvalues), which is found by solving E_n...E₂E₁AE₁^TE₂^T...E_n^T = A_RREF. In order to find D, take the column & row reductions one at a time - noting that you have to do AEE^T for every row manipulation E. (E is an elementary matrix s.t. det(E) = 1)

Using the method in post #4 (which is essentially Cholesky Decomposition, adapted to an indefinite matrix) you do not ever need to find eigenvalues, so is much faster and much more efficient, and possibly much more accurate as well.