AI 中文总结
研究任意对称矩阵Cholesky分解，通过类比正定矩阵与特定二次方程，利用半复数这种非结合代数实现分解，无需pivoting保持带状结构，对奇异矩阵可生成零空间参数化，为求解对称线性方程增添实用工具。
AI 中文摘要
正定矩阵使用Cholesky分解能最有效地进行因式分解。对于不定矩阵，Cholesky分解不存在，其他方法在实现数值稳定性和保持带状结构方面面临更大挑战。本文通过类比正定矩阵的要求与二次方程\(x^2 = c\)（\(c \leq 0\)）的解，证明了一种非结合代数——半复数，可用于计算任意对称矩阵的Cholesky分解。关键在于，半复数Cholesky分解在存在性和稳定性上无需 pivoting，能保持带状结构。对于奇异矩阵，它能生成零空间的参数化，并提供类似奇异值分解中截断近零方向的机会。半复数Cholesky分解可能是求解对称线性方程工具中实用的补充。
英文摘要
Positive-semidefinite matrices are most efficiently factored using the Cholesky decomposition. For indefinite matrices, the Cholesky factorization does not exist, and the alternatives face greater challenges in achieving numeric stability and preservation of banded structure. Here we pursue an analogy between the requirement for positive-semidefinite matrices and the solution of the quadratic equation x^2 = c for c <= 0. It is shown that a non-associative algebra, called the hemiplex numbers, allows the Cholesky factorization to be computed for arbitrary symmetric matrices. Crucially, the hemiplex Cholesky factorization does not require pivoting for its existence or stability, allowing it to preserve banded structure. For singular matrices it produces a parametrization of the null space, and provides opportunity for truncation of nearly-null directions in a manner similar to common usage of the singular value decomposition. The hemiplex Cholesky factorization may be a practically useful addition to the tools for solving symmetric linear equations.
Comments17 pages, 3 figures. Submitted to the SIAM Journal on Matrix Analysis and Applications. Software: https://github.com/timholy/HemiplexNumbers.jl and https://github.com/timholy/HemiplexFactorizations.jl
