Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices

Abstract

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.