A note on subset selection for matrices (Q630536)

The problem of selecting rows of a matrix \(X\) so that the resulting matrix \(A\) is ``as non-singular as possible'' is considered: given \(X\in\mathbb{R}^{m\times n}\) with \(n<m\) and rank\,\(X=n\), determine a permutation matrix \(P\in\mathbb{R}^{n\times n}\) so that \[ PX=\left[\begin{matrix} A\\B\end{matrix}\right],\;A\in\mathbb{R}^{k\times n},\;\text{rank\,}A=n,\;n\leq k<m. \] Upper bounds for \(\|A^+\|_F\) (where \(A^+\) is the Moore-Penrose inverse of \(A\)) and the singular values of \(A\) have been obtained by the authors in a previous paper. These results are improved in the present paper by providing a constructive derivation for the bounds of \(\|A^+\|_F\) and new lower bounds for \(\det A^TA\).