Let $F$ be an operator (in $\mathbb{R^3}$) whose eigenvalues are: $\lambda_1 =1$, $\lambda_2 = -1$, whose eigenspaces are: $V_1= \{(x_1, x_2, x_2): x_1-2x_2+x_3=0\}$ and $V_2=span\{(2,0,1)\}$.
- Find a positive-definite scalar product for which $F$ is symmetrical.
$F$ isn't simmetrycal for the inner product because its eigenvectors aren't orthogonal to each other: $(2,1,0)(2,0,1)\neq0$ for example. However, since $F$ is diagonalizable (we can see it with easy calculations), there is an orthogonal basis of eigenvectors for which $F$ is symmetrical. How to find a positive-definite scalar product in that basis?