Diagonalising the form by completing the square

Question

Write down the matrix A associated to the quadratic form $$3x^2 + 3y^2+3z^2-2xy-2xz-2yz$$ Classify the form as positive/negative definite/semidefinite, or indefinite. Diagonalise the form by completing the square.

I've worked out the matrix which is:

$$ \begin{pmatrix} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \\ \end{pmatrix} $$

How would I then complete the square to diagoanlise the form?

Any help is much appreciated :) Thank you!

Answer 1

In this case, it's quite simple:$$3x^2 + 3y^2+3z^2-2xy-2xz-2yz=(-x+y+z)^2+(x-y+z)^2+(x+y-z)^2.$$

Answer 2

There is the intuitive leap as in the other answer, but if you want a systematic method there is the beginning. $$3x^2 + 3y^2+3z^2-2xy-2xz-2yz$$ collect all $x$ terms $$=3(x^2 -\frac{2}{3}x(y+z))+ 3y^2+3z^2-2yz$$ complete the square $$=3(x^2 -\frac{2}{3}x(y+z)+\frac{(y+z)^2}{9}) -\frac{(y+z)^2}{3}+ 3y^2+3z^2-2yz$$ $$=3(x -\frac{1}{3}(y+z))^2-\frac{(y+z)^2}{3}+ 3y^2+3z^2-2yz$$ now factor the $y,z$ terms.

Answer 3

Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
$$ H = \left( \begin{array}{rrr} 3 & - 1 & - 1 \\ - 1 & 3 & - 1 \\ - 1 & - 1 & 3 \\ \end{array} \right) $$ $$ D_0 = H $$

$$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$

$$ H = \left( \begin{array}{rrr} 3 & - 1 & - 1 \\ - 1 & 3 & - 1 \\ - 1 & - 1 & 3 \\ \end{array} \right) $$

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$$ E_{1} = \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrr} 1 & - \frac{ 1 }{ 3 } & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrr} 3 & 0 & - 1 \\ 0 & \frac{ 8 }{ 3 } & - \frac{ 4 }{ 3 } \\ - 1 & - \frac{ 4 }{ 3 } & 3 \\ \end{array} \right) $$

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$$ E_{2} = \left( \begin{array}{rrr} 1 & 0 & \frac{ 1 }{ 3 } \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 3 } \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrr} 1 & - \frac{ 1 }{ 3 } & - \frac{ 1 }{ 3 } \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrr} 3 & 0 & 0 \\ 0 & \frac{ 8 }{ 3 } & - \frac{ 4 }{ 3 } \\ 0 & - \frac{ 4 }{ 3 } & \frac{ 8 }{ 3 } \\ \end{array} \right) $$

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$$ E_{3} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 2 } \\ 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrr} 1 & - \frac{ 1 }{ 3 } & - \frac{ 1 }{ 3 } \\ 0 & 1 & - \frac{ 1 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrr} 3 & 0 & 0 \\ 0 & \frac{ 8 }{ 3 } & 0 \\ 0 & 0 & 2 \\ \end{array} \right) $$

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$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ \frac{ 1 }{ 3 } & 1 & 0 \\ \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 3 & - 1 & - 1 \\ - 1 & 3 & - 1 \\ - 1 & - 1 & 3 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & \frac{ 1 }{ 3 } & \frac{ 1 }{ 2 } \\ 0 & 1 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 3 & 0 & 0 \\ 0 & \frac{ 8 }{ 3 } & 0 \\ 0 & 0 & 2 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ - \frac{ 1 }{ 3 } & 1 & 0 \\ - \frac{ 1 }{ 3 } & - \frac{ 1 }{ 2 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 3 & 0 & 0 \\ 0 & \frac{ 8 }{ 3 } & 0 \\ 0 & 0 & 2 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - \frac{ 1 }{ 3 } & - \frac{ 1 }{ 3 } \\ 0 & 1 & - \frac{ 1 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 3 & - 1 & - 1 \\ - 1 & 3 & - 1 \\ - 1 & - 1 & 3 \\ \end{array} \right) $$

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