[Paper Review] Using Quaternion-Valued Linear Algebra

Dominik Schulz, Reiner S. Thomä|arXiv (Cornell University)|Nov 29, 2013

Algebraic and Geometric Analysis9 references3 citations

TL;DR

This paper proposes a novel notation for quaternion-valued linear algebra that explicitly accounts for the non-commutativity of quaternion multiplication through left and right matrix products. By introducing left- and right-sided matrix multiplication operators, the authors enable a consistent, compact, and practical framework for quaternionic linear algebra, successfully applied to eigenvalue decomposition and the discrete quaternion Fourier transform (DQFT), achieving more concise and structured formulations than prior approaches.

ABSTRACT

Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear algebra over skew fields. To this end, we discuss ways of notation that account for the non-commutativity of the quaternion multiplication.

Motivation & Objective

• To address the lack of a standardized notation for linear algebra over skew fields like the quaternions.
• To resolve the fundamental challenge of non-commutative quaternion multiplication in matrix operations.
• To develop a consistent and practical notation for quaternionic matrix algebra that preserves order and enables clear formulation of operations.
• To demonstrate the utility of the proposed notation in key applications such as eigenvalue decomposition and the discrete quaternion Fourier transform (DQFT).
• To extend classical linear algebra tools—like Kronecker and Khatri-Rao products—to the quaternionic setting using the new notation.

Proposed method

• Proposes a left matrix product (·_L) and right matrix product (·_R) to explicitly encode multiplication order in quaternion matrix operations.
• Defines matrix multiplication as a sum over products where the order of quaternion multiplication is preserved: ∑[A]_m,k [B]_k,n for left multiplication and ∑[B]_k,n [A]_m,k for right multiplication.
• Introduces quaternion Fourier matrices F₁^(μ₁) and F₂^(μ₂) using exponential forms with pure unit quaternions μ₁ and μ₂, scaled by 1/√M and 1/√N.
• Applies the new notation to three variants of the DQFT: left-side, mixed, and right-side, each defined via ordered left and right matrix multiplications.
• Derives inverse DQFTs using conjugate transpose matrices (H) and maintains the same order-aware structure.
• Demonstrates that associativity is preserved only when μ₁ = μ₂, allowing arbitrary parenthesization in that special case.

Experimental results

Research questions

• RQ1 How can a consistent and unambiguous notation be developed for linear algebra over the skew field of quaternions, given the non-commutativity of multiplication?
• RQ2 Can the proposed left and right matrix product notation simplify and unify the formulation of fundamental quaternionic linear algebra operations?
• RQ3 How does the new notation improve the expression and analysis of the discrete quaternion Fourier transform (DQFT) compared to existing formulations?
• RQ4 To what extent can the new notation support the extension of classical matrix tools like Kronecker and Khatri-Rao products to the quaternion domain?
• RQ5 Under what conditions does the matrix product using the new notation become associative, and how does this affect transform inversion and decomposition?

Key findings

• The proposed left and right matrix product notation successfully resolves the ambiguity in quaternion matrix multiplication order, enabling a consistent framework for quaternionic linear algebra.
• The DQFT formulations using the new notation are more compact and structured, with three distinct variants (left, mixed, right) clearly defined via ordered matrix multiplications.
• The inverse DQFTs are derived using conjugate transpose matrices and maintain the same order-aware structure, ensuring invertibility.
• Associativity of the matrix product is achieved only when the same pure unit quaternion μ is used for both dimensions (μ₁ = μ₂), allowing flexible parenthesization.
• The notation enables a more natural and concise formulation of the eigenvalue decomposition over quaternions, improving clarity and reducing complexity.
• The extension of Kronecker and Khatri-Rao products to the quaternionic setting is shown to be feasible and meaningful within the proposed notation framework.