Define the Legendre transform $f^{*}(\mathbf{p})$ of a function $f(\mathbf{x})$ where $\mathbf{x} \in \mathbb{R}^{n}$.

Show that for $g(\mathbf{x})=\lambda f\left(\mathbf{x}-\mathbf{x}_{0}\right)-\mu$,

$$g^{*}(\mathbf{p})=\lambda f^{*}\left(\frac{\mathbf{p}}{\lambda}\right)+\mathbf{p}^{\mathbf{T}} \mathbf{x}_{0}+\mu$$

Show that for $f(\mathbf{x})=\frac{1}{2} \mathbf{x}^{\mathbf{T}} \mathbf{A} \mathbf{x}$ where $\mathbf{A}$ is a real, symmetric, invertible matrix with positive eigenvalues,

$$f^{*}(\mathbf{p})=\frac{1}{2} \mathbf{p}^{\mathbf{T}} \mathbf{A}^{-\mathbf{1}} \mathbf{p}$$