On the Cross-Shaped Matrices

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