Condition numbers of rectangular systems and bounds for generalized inverses
Motivated by the fact that the reciprocal of the condition number of a regular matrix is the smallest distance (in the matrix norm used to define the condition number) to a singular matrix, divided by the norm of the matrix, the author defines a condition number for arbitrary \(m\times n\)-matrices A as follows: \(Let\quad \alpha (A):=\min \{\| A^*x\| | x\in R(A),\| x\| =1\},\) which is positive for \(A\neq 0\), and define \(\kappa (A):=\| A\| \cdot \alpha (A)^{-1}\). This condition number enjoys the analogous property as mentioned above: \(\alpha (A)=\inf \{\| A-B\| | rank B<rank A\}.\) The author shows that \(\kappa\) (A) can be used for bounding the relative error for solvable linear systems with arbitrary matrices in terms of the relative error of the right-hand side. Another possible definition of a condition number for arbitrary matrices would be \({\hat \kappa}\)(A):\(=\| A\|\); \(\inf \{\| B\| | B\) linear and \(ABx=x\) for \(x\in R(A)\}\). If A is invertible or if the norms are Euclidean, then \(\kappa (A)={\hat \kappa}(A)\). The author shows that this is also true for \(n\times m\)-matrices with rank A\(=n<m\), if the \(l^ 1\)-norm is used. From deep results about projection constants in Banach spaces, the author derives that for all \(L^ p\)-norms with \(1<p\leq \infty\), \(p\neq 2\), \(\kappa\) (A) is in general much smaller than \({\hat \kappa}\)(A) in the following sense: For all such p, there exist sequences of integers \(n_ K\) and \(m_ K\) tending to \(\infty\) and \(n_ K\times m_ K\)-matrices \(A_ K\) such that for the respective condition numbers determined by the \(l^ p\)-norms, \(\kappa (A_ K)\) is bounded, but \({\hat \kappa}\)(A\({}_ K)\) tends to \(\infty\).
scientific article; zbMATH DE number 3994916