I'm having a really hard time solving this equation:
$q(x_1,x_2,x_3,x_4)=4x_1x_4 + 2x_2x_3$
I tried many different approaches and got to
$(4x_1^2+4x_1x_4+x_4^2-4x_1^2-x_4^2)+2x_2x_3= (2x_1+x_4)^2-4x_1^2-x_4^2+2x_2x_3$
After that if I try and complete $2x_2x_3$ to a square, I get stuck with too many monomials around.
I would really appreciate the help.
PS this is not some homework that I can't figure out, I'm trying to get ready for linear algebra 2 course, I try to study by myself before the semester starts.
Best Answer
If you are not feeling clever, see the algorithm at reference for linear algebra books that teach reverse Hermite method for symmetric matrices
When all off-diagonal coefficients are even, we may define $H$ to be HALF the Hessian matrix
The proposed matrix identity says $$\color{magenta}{ (x_1 + x_4)^2 - (x_1 - x_4)^2 + \frac{1}{2}(x_2+x_3)^2 - \frac{1}{2}(x_2-x_3)^2}$$
$$ P^T H P = D $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ 0 & 1 & 1 & 0 \\ 0 & \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 \\ 0 & 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 & - \frac{ 1 }{ 2 } \\ 1 & \frac{ 1 }{ 2 } & 0 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 2 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & - 1 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 1 \\ 0 & 0 & \frac{ 1 }{ 2 } & - 1 \\ \frac{ 1 }{ 2 } & 1 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 2 } \\ \end{array} \right) \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ - 1 & 0 & 0 & 1 \\ 0 & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 \\ 0 & 1 & - 1 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) $$
.....
Here is the whole sequence:
$$ H = \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$
$$ H = \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) $$
==============================================
$$ E_{1} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - 1 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrrr} 4 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) $$
==============================================
$$ E_{2} = \left( \begin{array}{rrrr} 1 & 0 & 0 & - \frac{ 1 }{ 2 } \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrrr} 1 & 0 & 0 & - \frac{ 1 }{ 2 } \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & \frac{ 1 }{ 2 } \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - 1 & 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ \end{array} \right) $$
==============================================
$$ E_{3} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & \frac{ 1 }{ 2 } & 0 & 0 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ - 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right) $$
==============================================
$$ E_{4} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{array} \right) $$ $$ P_{4} = \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & \frac{ 1 }{ 2 } & 0 & 0 \\ \end{array} \right) , \; \; \; Q_{4} = \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ - 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & - 1 & 0 \\ \end{array} \right) , \; \; \; D_{4} = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right) $$
==============================================
$$ E_{5} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - \frac{ 1 }{ 2 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{5} = \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 \\ 0 & 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 & - \frac{ 1 }{ 2 } \\ 1 & \frac{ 1 }{ 2 } & 0 & 0 \\ \end{array} \right) , \; \; \; Q_{5} = \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ - 1 & 0 & 0 & 1 \\ 0 & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 \\ 0 & 1 & - 1 & 0 \\ \end{array} \right) , \; \; \; D_{5} = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 2 } \\ \end{array} \right) $$
==============================================
$$ P^T H P = D $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ 0 & 1 & 1 & 0 \\ 0 & \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 \\ 0 & 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 & - \frac{ 1 }{ 2 } \\ 1 & \frac{ 1 }{ 2 } & 0 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 2 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & - 1 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 1 \\ 0 & 0 & \frac{ 1 }{ 2 } & - 1 \\ \frac{ 1 }{ 2 } & 1 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & - \frac{ 1 }{ 2 } \\ \end{array} \right) \left( \begin{array}{rrrr} \frac{ 1 }{ 2 } & 0 & 0 & \frac{ 1 }{ 2 } \\ - 1 & 0 & 0 & 1 \\ 0 & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 \\ 0 & 1 & - 1 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{array} \right) $$