标题： 关于交叉矩阵的注记 显示英文标题

标题： A note on the cross matrices

摘要： 一个交叉矩阵 $X$ 的非零元素只能位于主对角线和反对角线上，因此其稀疏模式呈十字形。 文中表明，$X$ 可以被分解为最多是秩二摄动的单位矩阵的乘积，并且可以通过对称排列化为具有 $2\times 2$ 个对角块的分块对角形式；如果 $n$ 是奇数，则还包含一个 $1\times 1$ 对角块。 置换相似性意味着 $X$ 的任何良好定义的解析函数仍然是一个交叉矩阵。 通过利用这些性质，推导出了行列式、逆矩阵以及特征多项式的显式公式。 此外，还证明了交叉矩阵的结构在矩阵分解下也可以保持不变，包括 LU 分解、QR 分解和奇异值分解（SVD）。 显示英文摘要

摘要： 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.