On the Cross-Shaped Matrices

Liu, Xiaobo (2023) On the Cross-Shaped Matrices. [MIMS Preprint]

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Abstract

A cross-shaped matrix $X\in\C^{n\times n}$ has nonzero elements located on the main diagonal and the anti-diagonal, so that the sparsity pattern has the shape of a cross. It is shown that $X$ can be factorized into products of identity-plus-rank-two matrices and can be symmetrically permuted to block diagonal form with $2\times 2$ diagonal blocks and, if $n$ is odd, a $1\times 1$ diagonal block. Exploiting these properties we derive explicit formulae for its determinant, inverse, and characteristic polynomial.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
Depositing User: Xiaobo Liu
Date Deposited: 20 Jan 2023 09:37
Last Modified: 20 Jan 2023 09:37
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2883

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