= Solution
Set $y=P^Tx$. Because $P$ is an <orthogonal matrix>, $x=Py$ and $x=0$ exactly when $y=0$. The preceding diagonalization gives
$$
x^TAx=y^TDy=y_1^2+11y_2^2\ge y_1^2+y_2^2=\|x\|_2^2.
$$
Consequently \b[$x^TAx\ge0$, with equality only when $x=0$]. The <matrix> is a <symmetric positive-definite matrix>, not merely positive semidefinite. In the original coordinates the same identity is
$$
x^TAx=\frac{(3x_1-x_2)^2+11(x_1+3x_2)^2}{10},
$$
which also makes positivity transparent.
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