Quadratic Forms

Dec 2009
225
3


What does it mean to write Q as a function of \(\displaystyle x_1\) and \(\displaystyle x_2\)? Is the following what they are asking for:

\(\displaystyle Q(x) = (x_1 x_2) \begin{pmatrix}4 & 3 \\ 3 & 5 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}\)

\(\displaystyle = 4x_1^2 + 5x^2_2 +6x_1x_2\)

Is this correct so far? And how do I need to "complete the square" to show that Q(x)>0? (I don't have any notes on this...)
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Yes, what you have so far, \(\displaystyle 4x_1^2+ 6x_1x_2+ 5x_2^2\), is correct.

And you "complete the square" just the way you learned many years ago in elementary algebra: To make \(\displaystyle 4x_1^2+ 6x_1x_2= 4(x_1^2+ (3/2)x_1x_2)\) a "perfect square", you have to add and subtract \(\displaystyle \left(\frac{\frac{3}{2}x_2}{2}\right)^2= \frac{9}{16}x_2^2\).

\(\displaystyle 4x_1^2+ 6x_1x_2+ \frac{9}{16}x_2^2- \frac{9}{16}x_2^2+ 5x_2^2= 4(x_1+ \frac{3}{4}x_2)^2+ \frac{71}{16}x_2^2\) which, as a sum of squares, is never negative.
 
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