Vrftjkry

I'm really lost at this:

Let q be the quadratic form on $mathbb{R}^3$ given by
$q(x_1, x_2, x_3) = x_1x_2 + 2x_1x_3 + x_3^2$.
Find an invertible linear operator U on $mathbb{R}^3$ such that
$(U^{dagger}q) (x_1, x_2, x_3) = x_1^2-x_2^2 + x_3^2$.

$(U^{dagger}q)(x_1, x_2, x_3) = q(U(x_1, x_2, x_3))$. I'm guessing that $U^{dagger}$ is like the pseudoinverse

Any advices?

The matrix identity that follows, divided through by $2,$ says
$$ (x+z)^2 - left(x - frac{y}{2} right)^2 + frac{1}{4} y^2 = z^2 + 2 zx + xy $$
This can be revised quickly to
$$ (x+z)^2 - left(x - frac{y}{2} right)^2 + left( frac{y}{2} right)^2 = z^2 + 2 zx + xy $$

$$ Q^T D Q = H $$
$$left(
begin{array}{rrr}
1 & 1 & 0 \
0 & - frac{ 1 }{ 2 } & 1 \
1 & 0 & 0 \
end{array}
right)
left(
begin{array}{rrr}
2 & 0 & 0 \
0 & - 2 & 0 \
0 & 0 & frac{ 1 }{ 2 } \
end{array}
right)
left(
begin{array}{rrr}
1 & 0 & 1 \
1 & - frac{ 1 }{ 2 } & 0 \
0 & 1 & 0 \
end{array}
right)
= left(
begin{array}{rrr}
0 & 1 & 2 \
1 & 0 & 0 \
2 & 0 & 2 \
end{array}
right)
$$

Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = left(
begin{array}{rrr}
0 & 1 & 2 \
1 & 0 & 0 \
2 & 0 & 2 \
end{array}
right)
$$
$$ D_0 = H $$
$$ E_j^T D_{j-1} E_j = D_j $$
$$ P_{j-1} E_j = P_j $$
$$ E_j^{-1} Q_{j-1} = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$

$$ H = left(
begin{array}{rrr}
0 & 1 & 2 \
1 & 0 & 0 \
2 & 0 & 2 \
end{array}
right)
$$

==============================================

$$ E_{1} = left(
begin{array}{rrr}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
end{array}
right)
$$
$$ P_{1} = left(
begin{array}{rrr}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
end{array}
right)
, ; ; ; Q_{1} = left(
begin{array}{rrr}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
end{array}
right)
, ; ; ; D_{1} = left(
begin{array}{rrr}
2 & 0 & 2 \
0 & 0 & 1 \
2 & 1 & 0 \
end{array}
right)
$$

==============================================

$$ E_{2} = left(
begin{array}{rrr}
1 & 0 & - 1 \
0 & 1 & 0 \
0 & 0 & 1 \
end{array}
right)
$$
$$ P_{2} = left(
begin{array}{rrr}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & - 1 \
end{array}
right)
, ; ; ; Q_{2} = left(
begin{array}{rrr}
1 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
end{array}
right)
, ; ; ; D_{2} = left(
begin{array}{rrr}
2 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & - 2 \
end{array}
right)
$$

==============================================

$$ E_{3} = left(
begin{array}{rrr}
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
end{array}
right)
$$
$$ P_{3} = left(
begin{array}{rrr}
0 & 1 & 0 \
0 & 0 & 1 \
1 & - 1 & 0 \
end{array}
right)
, ; ; ; Q_{3} = left(
begin{array}{rrr}
1 & 0 & 1 \
1 & 0 & 0 \
0 & 1 & 0 \
end{array}
right)
, ; ; ; D_{3} = left(
begin{array}{rrr}
2 & 0 & 0 \
0 & - 2 & 1 \
0 & 1 & 0 \
end{array}
right)
$$

==============================================

$$ E_{4} = left(
begin{array}{rrr}
1 & 0 & 0 \
0 & 1 & frac{ 1 }{ 2 } \
0 & 0 & 1 \
end{array}
right)
$$
$$ P_{4} = left(
begin{array}{rrr}
0 & 1 & frac{ 1 }{ 2 } \
0 & 0 & 1 \
1 & - 1 & - frac{ 1 }{ 2 } \
end{array}
right)
, ; ; ; Q_{4} = left(
begin{array}{rrr}
1 & 0 & 1 \
1 & - frac{ 1 }{ 2 } & 0 \
0 & 1 & 0 \
end{array}
right)
, ; ; ; D_{4} = left(
begin{array}{rrr}
2 & 0 & 0 \
0 & - 2 & 0 \
0 & 0 & frac{ 1 }{ 2 } \
end{array}
right)
$$

==============================================

$$ P^T H P = D $$
$$left(
begin{array}{rrr}
0 & 0 & 1 \
1 & 0 & - 1 \
frac{ 1 }{ 2 } & 1 & - frac{ 1 }{ 2 } \
end{array}
right)
left(
begin{array}{rrr}
0 & 1 & 2 \
1 & 0 & 0 \
2 & 0 & 2 \
end{array}
right)
left(
begin{array}{rrr}
0 & 1 & frac{ 1 }{ 2 } \
0 & 0 & 1 \
1 & - 1 & - frac{ 1 }{ 2 } \
end{array}
right)
= left(
begin{array}{rrr}
2 & 0 & 0 \
0 & - 2 & 0 \
0 & 0 & frac{ 1 }{ 2 } \
end{array}
right)
$$
$$ Q^T D Q = H $$
$$left(
begin{array}{rrr}
1 & 1 & 0 \
0 & - frac{ 1 }{ 2 } & 1 \
1 & 0 & 0 \
end{array}
right)
left(
begin{array}{rrr}
2 & 0 & 0 \
0 & - 2 & 0 \
0 & 0 & frac{ 1 }{ 2 } \
end{array}
right)
left(
begin{array}{rrr}
1 & 0 & 1 \
1 & - frac{ 1 }{ 2 } & 0 \
0 & 1 & 0 \
end{array}
right)
= left(
begin{array}{rrr}
0 & 1 & 2 \
1 & 0 & 0 \
2 & 0 & 2 \
end{array}
right)
$$