Diagonalization of a quadratic form

Question

I have to diagonalize the following quadratic form:

$$ Q(x,y,z,t)=x^2+y^2+2xy+2xz+2xt+2yz $$

I want to complete the squares, so I observe that the first three terms are the square $(x+y)^2$. Then I sum and subtract the quantities $2x^2$, $2z^2$, $y^2$ and $t^2$. Now, how do I have to continue? Which is the associated matrix?

Answer 1

Consume all the terms containing $x$ first ... etc ... \begin{eqnarray*} (x+y+z+t)^2-(y+z+t)^2+(y+z)^2-z^2 \end{eqnarray*}

Edit: \begin{eqnarray*} \begin{pmatrix}1 & 1 & 1&1 \\0 & 1 & 1&1 \\0 & 1& 1&0 \\0&0&1&0 \\\end{pmatrix}^{T} \begin{pmatrix}1 & 0 & 0&0 \\0 & -1 & 0&0 \\0 & 0& 1&0 \\0&0&0&-1 \\\end{pmatrix} \begin{pmatrix}1 & 1 & 1&1 \\0 & 1 & 1&1 \\0 & 1& 1&0 \\0&0&1&0 \\\end{pmatrix}= \begin{pmatrix}1 & 1 & 1&1 \\1 & 1 & 1&0 \\1 & 1& 0&0 \\1&0&0&0 \\\end{pmatrix} \end{eqnarray*}