I am new to Mathematica and am trying to check if the following matrix is positive definite with the program. The answer is supposed to be yes because $x > 0$ and $y > 0$ but I don't know how to add this into the function.
I know it is positive definite but I'm trying to use Mathematica to show that it is positive definite.
On
If you are using Mathematica, copy your equation display right click, copy as LaTeX and paste/post it here between $$\$\$\text{hello}\$\$ $$
$$\text{Eigenvalues}\left[\left( \begin{array}{cc} 8 \left(7 x^6+3 x^2 y\right) & 8 x^3 \\ 8 x^3 & 2 \\ \end{array} \right)\right]$$
or copy the original straight input with and indent or between {} in the tool bar of the input window
Eigenvalues[({{8 (7 x^6 + 3 x^2 y), 8 x ^3}, {8 x ^3, 2} })]
$$ 28 x^6+12 x^2 y-\sqrt{784 x^{12}+672 x^8 y+8 x^6+144 x^4 y^2-24 x^2 y+1}+1$$, $$28 x^6+12 x^2 y+\sqrt{784 x^{12}+672 x^8 y+8 x^6+144 x^4 y^2-24 x^2 y+1}+1 $$
$$ \det \left| \begin{array}{cc} 8 \left(7 x^6+3 x^2 y\right) & 8 x^3 \\ 8 x^3 & 2 \\ \end{array} \right| = 48 x^6 + 48 x^2 y $$
The $y$ term makes the determinant indefinite
Yes, it is positive definite. One of several equivalent necessary and sufficient conditions is $q_{22}>0$ and $\det Q>0$.Note that $\det Q=48(x^6+x^2y)$.