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A generalization of the Moore-Penrose inverse related to matrix subspaces of ${ℂ}^{n×m}$. (English) Zbl 1187.15008
Let $A\in {ℂ}^{m×n}$ and let $S$ be a linear subspace of ${ℂ}^{n×m}$. The authors define the left $S$-Moore-Penrose inverse of $A$ to be the minimum Frobenius norm solution to the matrix minimization problem ${min}_{M\in S}{\parallel MA-{I}_{n}\parallel }_{F}$, where ${I}_{n}$ denotes the identity matrix of order $n$ and ${\parallel ·\parallel }_{F}$ stands for the matrix Frobenius norm, and similarly define the right $S$-Moore-Penrose inverse of $A$. This is clearly a natural generalization of the classical Moore-Penrose inverse. They provide an explicit expression based on the singular value decomposition of the matrix $A$ as well as an alternative expression for the full rank case, in terms of the orthogonal complements with respect to the Frobenius inner product. These results are applied to the preconditioning of linear systems based on Frobenius norm minimization and to the linearly constrained linear least squares problem.
##### MSC:
 15A09 Matrix inversion, generalized inverses 65F08 Preconditioners for iterative methods
##### References:
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