[Paper Review] Matrix Theory over the Complex Quaternion Algebra

Yongge Tian|ArXiv.org|Apr 1, 2000

Algebraic and Geometric AnalysisMathematics7 references22 citations

TL;DR

This paper establishes a universal similarity factorization identity that links complex quaternion matrices to their 2×2 complex matrix representations via a fixed invertible matrix Q, enabling a consistent framework for defining generalized inverses, eigenvalues, determinants, and other matrix-theoretic concepts over the complex quaternion algebra. The key contribution is a canonical isomorphism between the complex quaternion algebra and the matrix algebra through a universal similarity transformation, which facilitates the extension of classical matrix theory to noncommutative quaternions.

ABSTRACT

We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex quaternion matrices, and so on.

Motivation & Objective

• To develop a systematic matrix theory over the complex quaternion algebra, which is noncommutative and lacks standard matrix tools.
• To resolve foundational challenges in defining matrix operations—such as inverses, eigenvalues, and determinants—on complex quaternion matrices due to noncommutativity.
• To establish a universal similarity factorization that links complex quaternions to 2×2 complex matrices, enabling the transfer of classical matrix theory results.
• To define a central determinant for complex quaternion matrices using their faithful matrix representations, ensuring consistency with multiplicative and identity properties.
• To extend classical results like the Cayley-Hamilton theorem to the complex quaternion setting via the new determinant definition.

Proposed method

• Introduces a universal similarity factorization: $ Q^{-1} \text{diag}(a, a) Q = \psi(a) $, where $ Q $ is a fixed 2×2 matrix over $ \mathbb{Q} $, independent of $ a $, and $ \psi(a) $ is the standard 2×2 complex matrix representation of $ a \in \mathbb{Q} $.
• Extends the factorization to $ m \times n $ matrices over $ \mathbb{Q} $, using block-diagonal and block-matrix constructions.
• Defines the central determinant of a complex quaternion matrix $ A \in \mathbb{Q}^{n \times n} $ as $ |A|_c = |\Psi(A)| $, the ordinary determinant of its complex matrix representation.
• Applies the universal similarity factorization to derive properties of generalized inverses, eigenvalues, and eigenvectors in the complex quaternion setting.
• Uses the isomorphism between $ \mathbb{Q} $ and $ \mathbb{C}^{2 \times 2} $ to lift standard matrix identities, such as the Cayley-Hamilton theorem, to the quaternion algebra.
• Applies the factorization to define and analyze similarity, Hermitian conjugation, and norms in the complex quaternion framework.

Experimental results

Research questions

• RQ1 How can matrix operations such as inverses and eigenvalues be consistently defined over the noncommutative complex quaternion algebra?
• RQ2 Can a universal similarity factorization be constructed that relates complex quaternion matrices to their 2×2 complex matrix representations without dependence on the matrix entries?
• RQ3 What is the appropriate definition of the determinant for complex quaternion matrices that preserves multiplicative and identity properties?
• RQ4 How can the Cayley-Hamilton theorem be extended to complex quaternion matrices using a well-defined determinant?
• RQ5 What are the structural properties of generalized inverses, similarity, and norms in the context of complex quaternion matrices?

Key findings

• A universal similarity factorization exists: $ Q^{-1} \text{diag}(a, a) Q = \psi(a) $, where $ Q $ is independent of $ a $, providing a canonical isomorphism between $ \mathbb{Q} $ and $ \mathbb{C}^{2 \times 2} $.
• The central determinant $ |A|_c = |\Psi(A)| $ is proposed as the natural determinant for $ A \in \mathbb{Q}^{n \times n} $, satisfying $ |AB|_c = |A|_c |B|_c $ and $ |A|_c \neq 0 $ iff $ A $ is invertible.
• The Cayley-Hamilton theorem holds for complex quaternion matrices: $ p_A(A) = 0 $, where $ p_A(\lambda) = |\lambda I_{2n} - \Psi(A)| $, a complex polynomial of degree $ 2n $.
• For diagonal matrices $ A = \text{diag}(a_{11}, \dots, a_{nn}) $, the central determinant satisfies $ |A|_c = n(a_{11}) \cdots n(a_{nn}) $, where $ n(a) = a_0^2 + a_1^2 + a_2^2 + a_3^2 $.
• The central determinant satisfies $ |\lambda A|_c = \lambda^{2n} |A|_c $ for $ \lambda \in \mathbb{C} $, and $ |\mu A|_c = n^2(\mu) |A|_c $ for $ \mu \in \mathbb{Q} $, with $ n(\mu) $ the weak norm.
• The central determinant is invariant under similarity: if $ A \sim B $, then $ |A|_c = |B|_c $, ensuring consistency across equivalent representations.