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.

翻译：交叉矩阵$X$的非零元素仅位于主对角线和反对角线上，其稀疏模式呈交叉形状。研究表明，$X$可分解为至多是单位矩阵的秩二修正矩阵的乘积，且可通过对称置换转化为块对角形式——其对角块为$2\times 2$子矩阵，当$n$为奇数时还包含一个$1\times 1$对角块。该置换相似性意味着$X$的任意良定义解析函数仍保持交叉矩阵结构。基于这些性质，本文推导了行列式、逆矩阵及特征多项式的显式公式。研究还表明，在LU分解、QR分解和奇异值分解等矩阵分解过程中，交叉矩阵的结构特性得以保持。