Express a quadratic form as a sum of squares using Schur complements I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows:

Consider $2x^2 + 2xy + 2y^2 + z^2 + 2xz$. Write the symmetric matrix representing this quadratic form. Now, express this as a sum of squares by using this symmetric matrix and Schur complements.

I determined the symmetric matrix representation as:
$$\begin{bmatrix}
        2 & 1 & 1 \\
        1 & 2 & 1 \\
        1 & 0 & 1 \\
        \end{bmatrix} $$
And that's as far as I've gotten. Any help would be much appreciated.
 A: I think what the question means is to apply the $L^TDL$ decomposition that produces Schur complements, rather than to use Schur complements themselves. In other words, you are expected to apply recursively the decomposition
$$
\pmatrix{A&B\\ B^T&C}=
\pmatrix{I&BC^{-1}\\ 0&I}
\pmatrix{A-BC^{-1}B^T&0\\ 0&C}
\pmatrix{I&0\\ C^{-1}B^T&I}
$$
to diagonalise a symmetric matrix by matrix congruence. In your case, apply $L^TDL$ decomposition once, we get
$$
\left(\begin{array}{cc|c}
2&1&1 \\
1&2&0 \\
\hline
1&0&1
\end{array}\right)
=\pmatrix{1&0&1\\ 0&1&0\\ 0&0&1}
\pmatrix{1&1&0\\ 1&2&0\\ 0&0&1}
\pmatrix{1&0&0\\ 0&1&0\\ 1&0&1}.
$$
Apply the decomposition once more to the submatrix
$
\left[\begin{array}{c|c}
1&1\\
\hline
1&2
\end{array}\right],
$
we get
\begin{align}
\pmatrix{
2&1&1\\
1&2&0\\
1&0&1}
=\pmatrix{1&0&1\\ 0&1&0\\ 0&0&1}
\pmatrix{1&\frac12&0\\ 0&1&0\\ 0&0&1}
\pmatrix{\frac12&0&0\\ 0&2&0\\ 0&0&1}
\pmatrix{1&0&0\\ \frac12&1&0\\ 0&0&1}
\pmatrix{1&0&0\\ 0&1&0\\ 1&0&1}.
\end{align}
Therefore, your matrix can be written as $L^TDL$ for some invertible matrix $L$, where $D=\operatorname{diag}(\frac12,2,1)$. Put $(u,v,w)=(x,y,z)L^T$, your quadratic form becomes $\frac12u^2+2v^2+w^2$, which is a sum of squares.
A: This problem is basically just orthogonal diagonalization (which is always possible with a real symmetric matrix)...and the question specifies the method you need to use. The Schur complements method can be applied as follow (I'm using a theorem from the book Matrices and Linear Transformations from CG Cullen): You need to find a unit vector \begin{equation}X = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix}x_1 \\ Y \end{bmatrix} \end{equation} with $x_1 \neq -1$. Then \begin{equation} P = \begin{bmatrix} x_1 & Y^T \\ Y & -I+\frac{1}{1+x_1}YY^T \end{bmatrix} \end{equation} is an orthogonal matrix.
So basically, what you need to do: Find the eigenvalues (just using the characteristic equation as usual), then determine one eigenvector associated with a particular eigenvalue of your choice. Normalize this eigenvector, and then apply the formula above. The matrix $P$ is then your change of basis to the new coordinate system, and the eigenvalues will be the coefficients of the squares in your new sum of squares equation. So you'll have \begin{equation} \lambda_1u^2+\lambda_2v^2+\lambda_3w^2\end{equation} where $u,v,w$ are the variables corresponding to the new coordinate system defined by the columns of $P$.
