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Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin’s $$\sin (\theta)$$ theorem. (English) Zbl 07329854
Summary: For an $$m \times n$$ matrix $$A$$, the mathematical property that the rank of $$A$$ is equal to $$r$$ for $$0< r < \min (m,n)$$ is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency $$k$$ in a mathematically rigorous way and using only floating-point arithmetic. Given an integer $$k$$ and a real or complex matrix $$A$$, square or rectangular, we first present a verification algorithm to compute a narrow interval matrix $$\varDelta$$ with the property that there exists a matrix within $$A-\varDelta$$ with at least rank deficiency $$k$$. Subsequently, we extend this algorithm for computing an inclusion for a specific perturbation $$E$$ with that property but also a minimal distance with respect to any unitarily invariant norm. For this purpose, we generalize Wedin’s $$\sin (\theta)$$ theorem by removing its orthogonality assumption. The corresponding result is the singular vector space counterpart to Davis and Kahan’s generalized $$\sin (\theta)$$ theorem for eigenspaces. The presented verification methods use only standard floating-point operations and are completely rigorous including all possible rounding errors and/or data dependencies.
##### MSC:
 65F99 Numerical linear algebra 15A03 Vector spaces, linear dependence, rank, lineability