Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913309839589376 |
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| author | Kędzierski, Oskar |
| author_facet | Kędzierski, Oskar |
| contents | We give a constructive characterization of matrices satisfying the reverse-order law for the Moore--Penrose pseudoinverse. In particular, for a given matrix $A$ we construct another matrix $B$, of arbitrary compatible size and chosen rank, in terms of the right singular vectors of $A$, such that the reverse order law for $AB$ is satisfied. Moreover, we show that any matrix satisfying this law comes from a similar construction. As a consequence, several equivalent conditions to $B^+ A^+$ being a pseudoinverse of $AB$ are given, for example $\mathcal{C}(A^*AB)=\mathcal{C}(BB^*A^*)$ or $B\left(AB\right)^+A$ being an orthogonal projection. In addition, we parameterize all possible SVD decompositions of a fixed matrix and give Greville-like equivalent conditions for $B^+A^+$ being a $\{1,2\}$-,$\{1,2,3\}$- and $\{1,2,4\}$-inverse of $AB$, with a geometric insight in terms of the principal angles between $\mathcal{C}(A^*)$ and $\mathcal{C}(B)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse Kędzierski, Oskar Numerical Analysis 15A09, 15-04, 15A24, 15A18, 47A05, 65F20 We give a constructive characterization of matrices satisfying the reverse-order law for the Moore--Penrose pseudoinverse. In particular, for a given matrix $A$ we construct another matrix $B$, of arbitrary compatible size and chosen rank, in terms of the right singular vectors of $A$, such that the reverse order law for $AB$ is satisfied. Moreover, we show that any matrix satisfying this law comes from a similar construction. As a consequence, several equivalent conditions to $B^+ A^+$ being a pseudoinverse of $AB$ are given, for example $\mathcal{C}(A^*AB)=\mathcal{C}(BB^*A^*)$ or $B\left(AB\right)^+A$ being an orthogonal projection. In addition, we parameterize all possible SVD decompositions of a fixed matrix and give Greville-like equivalent conditions for $B^+A^+$ being a $\{1,2\}$-,$\{1,2,3\}$- and $\{1,2,4\}$-inverse of $AB$, with a geometric insight in terms of the principal angles between $\mathcal{C}(A^*)$ and $\mathcal{C}(B)$. |
| title | Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse |
| topic | Numerical Analysis 15A09, 15-04, 15A24, 15A18, 47A05, 65F20 |
| url | https://arxiv.org/abs/2404.02843 |