Optimal ellipsoids and decomposition of positive definite matrices
Let \(A\) be a positive definite matrix. The aim of the paper is to give a characterization of the unique diagonal matrix \(D\) with \(D\geq A\) with the smallest determinant. Such a matrix \(D\) is the optimizer if and only if (a) \(D\geq A\) and \(D-A\) is singular, and (b) there is an \(n\times n\) matrix \(X\geq 0\) with \((D-A)X=0\) and \(\text{diag} (X)=D^{-1}\). This problem has the following geometrical interpretation. Given an ellipsoid \(\mathcal A\) in \(\mathbb R^n\), find the ellipsoid \(\mathcal D\), contained in \(\mathcal A\), with principal axes parallel to the coordinate axes, of the largest volume. Some other consequences of this optimization problem are also derived.
scientific article; zbMATH DE number 5146751