A note on the cross matrices
Xiaobo Liu
TL;DR
The paper introduces cross matrices with sparsity on the main and anti-diagonals and derives two main structural results: a factorization into rank-two perturbations of the identity and a permutation-based block-diagonalization into $2\times 2$ blocks (with a possible $1\times 1$ center). These enable explicit expressions for the determinant, inverse, and eigenvalues, and show that cross matrices are closed under polynomial and analytic functions. Moreover, standard factorizations such as LU, QR, SVD, and polar decomposition can be performed while preserving the cross structure, yielding a structure-preserving toolkit for analysis and computation of cross matrices. These results have potential applications in symmetric eigenvalue problems and related numerical linear algebra tasks where sparsity and structure are advantageous.
Abstract
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.