Completing the square using a matrix in quadratic form

[PhizKid]

If I have a symmetric matrix for a given quadratic form, how do I complete the square using this matrix to put the quadratic as the sum and difference of squares?

 

[HallsofIvy]

There is NOT necessarily possible to write a quadratic form as a "sum and difference". It is always possible to write a symmetric matrix as either a difference of squares (which can be written as sum and difference) or as a su of squares.

The symmetric matrix $\displaystyle A= \begin{bmatrix}a & b \\ b & c \end{bmatrix}$ gives the quadratic form $\displaystyle X^TAX= \begin{bmatrix}x & y \end{bmatrix}\begin{bmatrix}a & b \\ b & c \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= ax^2+ 2bxy+ cy^2$. Now, because this matrix is symmetric, we know it is "diagonalizable". There exist an invertible matrix, P, and a diagonal matrix, D, such that $\displaystyle A= P^{-1}DP$ so that we can write $\displaystyle X^TAX= X^T(P^{-1}DP)X= (X^TP^{-1})D(PX)= Y^TDY$ where Y= PX.

That then gives $\displaystyle \begin{bmatrix}y_1 & y_2\end{bmatrix}\begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}\begin{bmatrix}y_1 \\ y_2 \end{bmatrix}= a_1y_1^2+ a_2y_2^2$. Whether that is a difference of squares or a sum of squares depends upon the signs of $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$.

Of course, $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ are the eigenvalues of A and the matrix, P, has the eigenvectors of A as columns.

 

[daon2]

You need that A is orthogonally diagonalizable for the above to work, which again is implied by A being symmetric. You must make P orthogonal in order for $\displaystyle (X^TP^{-1})^T= PX$