Let $x\in {\mathbb{R}}^{n}$ and let $P$ be a $n\times n$ positive definite symmetrix matrix. It is known that the maximum of

$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& {x}^{T}x\le 1\end{array}$

is ${\lambda}_{\text{max}}(P)$, the largest eigenvalue of $P$. Now consider the following problem

$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& (x-a{)}^{T}(x-a)\le 1\end{array}$

where $a\in {\mathbb{R}}^{n}$ is known. What is the analytical solution?

$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& {x}^{T}x\le 1\end{array}$

is ${\lambda}_{\text{max}}(P)$, the largest eigenvalue of $P$. Now consider the following problem

$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& (x-a{)}^{T}(x-a)\le 1\end{array}$

where $a\in {\mathbb{R}}^{n}$ is known. What is the analytical solution?