  We study a preconditioner for a Hermitian positive definite linear system,
which is obtained as the solution of a matrix nearness problem based on the
Bregman log determinant divergence. The preconditioner is of the form of a
Hermitian positive definite matrix plus a low-rank matrix. For this choice of
structure, the generalised eigenvalues of the preconditioned matrix are easily
calculated, and we show under which conditions the preconditioner minimises the
$\ell_2$ condition number of the preconditioned matrix. We develop practical
numerical approximations of the preconditioner based on the randomised singular
value decomposition (SVD) and the Nystr\"om approximation and provide
corresponding approximation results. Furthermore, we prove that the Nystr\"om
approximation is in fact also a matrix approximation in a range-restricted
Bregman divergence and establish several connections between this divergence
and matrix nearness problems in different measures. Numerical examples are
provided to support the theoretical results.

Numerical Analysis;Numerical Analysis