I am working on the problem of finding the (smallest) hypersphere that encloses a known ellipsoid, with possible equality on the ellipsoid's largest semi-axes, without computing any eigenvalues. I am attempting to set about this as follows:
Let $$ (\mathbf{x}-\mathbf{x}_0)^\intercal \mathbf{A}(\mathbf{x}-\mathbf{x}_0) = 1 $$ describe the ellipsoid, $E$, concerned. To be explicit, $\mathbf{A}$ is symmetric positive definite and nonsingular. Then by observing that $$ r_0 := r(\mathbf{A}^{-1}) $$ where $r(\mathbf{A}^{-1})$ is the spectral radius of $\mathbf{A}^{-1}$, $\sqrt{r_0}$ represents the length of the largest semi-axis of $E$. Therefore the hypersphere $C$ described by $$ (\mathbf{x}-\mathbf{x}_0)^\intercal(\mathbf{x}-\mathbf{x}_0) = r_0 $$ is exactly the smallest hypersphere that encloses $E$ with equality occurring at the end-point of each semi-axis of $E$ having $r_0$ as its length.
Now, to avoid computing any eigenvalues I am considering the formula of Gelfand which states that for any matrix $\mathbf{M}$ and consistent matrix norm $\|\cdot\|$, $$ r(\mathbf{M}) \leq \|\mathbf{M}^k\|^{\frac{1}{k}} =: n_k(\mathbf{M}) $$ for all $k$. Therefore any hypersphere $C_k$ described by $$ (\mathbf{x}-\mathbf{x}_0)^\intercal(\mathbf{x}-\mathbf{x}_0) = n_k(\mathbf{A}^{-1}) $$ encloses $E$, with $C_k = C$ in the case of Gelfand's limit $$ r_0 = \lim_{k\rightarrow\infty} \left\| \left(\mathbf{A}^{-1}\right)^k \right\|^{\frac{1}{k}}. $$
My questions therefore are:
- Is this approach a valid one as described?
- If so, what knowledge is there on the rate of convergence of Gelfand's limit, or better still, the error bounds, in terms of $k$, $\|\cdot\|$ and the form of $\mathbf{A}$? And therefore,
- Is there an optimal matrix norm that achieves minimal error bound given the particular form of $\mathbf{A}$?
