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A Note on the Cross Matrices by Liu, Xiaobo is a scholarly article available to read on EtoBox.

What is A Note on the Cross Matrices about?

A cross matrix $X$ can have nonzero elements located only 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 matrices that are at most rank-two perturbations to the identity matrix 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. The permutation similarity implies that any well-defined analytic function of $X$ remains a cross matrix. By exploiting these properties, explicit formulae for the determinant, inverse, and characteristic polynomial are derived. It is also shown that the structure of cross matrix can be preserved under matrix factorizations, including the LU, QR, and SVD decompositions.

Author

Liu, Xiaobo

Published

2025

Language

EN