Effective Matrix Methods in Commutative Domains

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

💡 Research Summary

The paper surveys and extends effective matrix algorithms for a commutative domain R equipped with an exact‑division routine. It treats six fundamental linear‑algebra tasks—matrix multiplication, solving linear systems over the fraction field K, solving linear systems directly in R, computing the adjoint (classical adjugate) matrix, determining the determinant, and obtaining the characteristic polynomial. For each task the author evaluates the asymptotic cost in terms of the number of ring multiplications and exact divisions, using the standard notation O(n^β) for matrix multiplication (β = 3 for the classical algorithm, β = log₂7 for Strassen, and β < 2 for the best known algorithms).

The first major contribution is a new binary factorization method for the adjoint matrix. By recursively splitting an n = 2^p matrix into blocks of size 2^{p‑k‑1} and applying a Strassen‑type multiplication at each level, the algorithm computes the adjoint with the same asymptotic complexity as matrix multiplication, namely O(n^β). The detailed analysis shows that the total number of multiplication‑division operations is C(n) = 3·α·n^β / (1 − 2^{1‑β}), where α is the constant hidden in the multiplication cost M(n)=α n^β. For β = 3 this yields a constant factor of 4, and for β = log₂7 a factor of about 4.2, substantially improving on the previously best O(n³√n log n log log n) bound.

The second major contribution is a randomized algorithm for solving linear systems in a principal‑ideal domain (PID) such as ℤ or F