A norm bound for projections with complex weights
Let \(\mu\geqslant 0\), \(C_\mu=\{z\in{\mathbb C}\mid |\text{Im }z|\leqslant\text{Re }z\), \(z\neq 0\}\), and let \({\mathcal D}_\mu=\{\text{diag}(a_1,\dots,a_n)\in M_m({\mathbb C})\mid a_1,\dots,a_m\in C_\mu\}\). Then for a rank \(n\) matrix \(A\in M_{m,n}({\mathbb R})\) there exist numbers \(\chi_{A,\mu},\overline\chi_{A,\mu}\in{\mathbb R}\) such that \(\|(A^tDA)^{-1}A^tD\|\leqslant \chi_{A,\mu}\) and \(\|A(A^tDA)^{-1}A^tD\|\leqslant \overline \chi_{A,\mu}\). For \(\mu=0\) these estimates are known as Dikin-Stewart-Todd norm bounds for weighted projections in the case of an ill-conditioned real positive-definite weight matrix [see \textit{A. L. Forsgren}, SIAM J. Matrix Anal. Appl. 17, No. 4, 763-788 (1996; Zbl 0885.65046)].
ill-conditioned real positive-definite weight matrix
scientific article; zbMATH DE number 1441331