Condition number related with \(W\)-weighted Drazin inverse and singular linear systems
If \(A\) and \(W\) are arbitrary \(m \times n\) and \(n \times m\) complex matrices, respectively, then \(X=[(AW)^D]^2A\) is a unique solution of the following equations: \[ (AW)^{k+1}XW=(AW)^k, \qquad XWAWX=X, \qquad AWX=XWA, \] where \(k\) as the index of \(W\) is the smallest nonnegative integer for which \(\text{ rank}[(AW)^k]=\text{ rank}[(AW)^{k+1}]\). The matrix \(X\) is called the \(W\)-weighted Drazin inverse of \(A\) and is written as \(X=A_{d,w}\). In this paper, the so-called \(PQ\)-norm is considered and the normwise relative condition number of the \(W\)-weighted Drazin inverse is discussed. Using the \(PQ\)-norm the previous results on the condition number regarding the unique solution \(x=A_{d,w}b\) of the singular linear systems \(WAWx=b, b \in R((WA)^{k_2}), x \in R((AW)^{k_1})\) are extended.
scientific article; zbMATH DE number 2136183