Domain for which this matrix is positive definite

Question

What is the domain for which this matrix is positive definite? $$\left(\begin{array}{cc} 12x^2 & 1 \\ 1 & 2 \\ \end{array}\right)$$

I'm trying to figure this out. I know the answer is that $12x^2 = 1/2$ from doing guess and check using Cholesky decomposition in Matlab. Thanks for any help.

Answer 1

Via Sylvester's criterion, you need $12x^2 \gt 0$ and $24x^2-1\gt 0$. From the first condition we have $|x| \gt 0$ and from the second we have $|x|\gt \frac 1{\sqrt{24}}$. The intersection of these two sets is $\{x\ {\big |}\ |x|\gt \frac 1{\sqrt{24}}\}$.

Answer 2

Consider the characteristic polynomial $$ p(l)=(l-12x^2)(l-2)-1=l^2-l(12x^2+2)+24x^2-1. $$ You can verify that the discriminant ($8 (1 - 6 x^2 + 18 x^4)$) is strictly positive so $p$ carries 2 real roots. They sum to $12x^2+2$ which is always positive and they multiply to $24x^2-1$. Thus, to have both positive roots (which is equivalent to your matrix being positive definite), it suffices to have $24x^2-1>0$. This is equivalent to $x>\frac{1}{\sqrt{24}}$ or $x<-\frac{1}{\sqrt{24}}$.