Strong DMP inverse

Sanzhang Xu; Zhengyang Shan; Wenqi Li; Xiaofei Cao; Ber-Lin Yu; Sanzhang Xu; Zhengyang Shan; Wenqi Li; Xiaofei Cao; Ber-Lin Yu

doi:10.3934/math.20251319

AIMS Mathematics

2025, Volume 10, Issue 12: 30017-30028. doi: 10.3934/math.20251319

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Research article Topical Sections

Strong DMP inverse

Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian 223003, China

Received: 02 October 2025 Revised: 12 December 2025 Accepted: 16 December 2025 Published: 22 December 2025
MSC : 15A09

A new-type matrix inverse based on the Hartwig-Spindelböck decomposition was investigated, which is related to the DMP inverse, is the abbreviation of the Drazin Moore-Penrose inverse, here we call this generalized inverse as the strong DMP inverse. We established the relationships between this new-type inverse and other matrix generalized inverses. We proved that the strong DMP inverse of a square complex matrix is a special type of outer inverse, where the restricted column space is $ \mathcal{R}(A^{k}) $ and restricted null space is $ \mathcal{N}(AA^{D}\alpha_{A}) $, where $ \alpha_{A} = AA^{\dagger}+(A^{\dagger})^{\ast}(E_{n}-AA^{\dagger}) $. The one-sided strong DMP inverses were introduced from the perspectives of the column space, null space, and index of a matrix. The general expressions of the left (right) strong DMP inverse of $ A $ were given. We answered the question of when the left strong DMP inverse is consistent with the right strong DMP inverse. Based on the relationship between the left strong DMP inverse and the right DMP inverse, the necessary and sufficient conditions for the existence of the strong DMP inverse were given.
- DMP inverse,
- Hartwig-Spindelböck decomposition,
- one-sided strong DMP inverse,
- column space,
- null space
Citation: Sanzhang Xu, Zhengyang Shan, Wenqi Li, Xiaofei Cao, Ber-Lin Yu. Strong DMP inverse[J]. AIMS Mathematics, 2025, 10(12): 30017-30028. doi: 10.3934/math.20251319

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Abstract

A new-type matrix inverse based on the Hartwig-Spindelböck decomposition was investigated, which is related to the DMP inverse, is the abbreviation of the Drazin Moore-Penrose inverse, here we call this generalized inverse as the strong DMP inverse. We established the relationships between this new-type inverse and other matrix generalized inverses. We proved that the strong DMP inverse of a square complex matrix is a special type of outer inverse, where the restricted column space is $ \mathcal{R}(A^{k}) $ and restricted null space is $ \mathcal{N}(AA^{D}\alpha_{A}) $, where $ \alpha_{A} = AA^{\dagger}+(A^{\dagger})^{\ast}(E_{n}-AA^{\dagger}) $. The one-sided strong DMP inverses were introduced from the perspectives of the column space, null space, and index of a matrix. The general expressions of the left (right) strong DMP inverse of $ A $ were given. We answered the question of when the left strong DMP inverse is consistent with the right strong DMP inverse. Based on the relationship between the left strong DMP inverse and the right DMP inverse, the necessary and sufficient conditions for the existence of the strong DMP inverse were given.

References

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