In linear algebra, the quaternionic eigenvalue problem is the problem of determining the eigenvalues and eigenvectors of a matrix with quaternionic entries. Unlike the classical eigenvalue problem over the complex numbers, where multiplication is commutative, two distinct problems can be considered: the right eigenvalue problem and the left eigenvalue problem.
History The study of eigenvalues of quaternionic matrices began in the late 1940s with the work of H. C. Lee, who established quaternionic analogues of several classical results of matrix theory, including a version of Shcur's triangularization theorem. Afterwards, N. A. Wiegmann established the existence of a Jordan canonical form for quaternionic matrices. Most of this early work focused on the right eigenvalue problem. In the 1980s, R. M. W. Wood proved the existence of left eigenvalues. Subsequent work by W. So, L. Huang and others developed computational methods, localization theorems and classifications of left eigenvalues for low-dimensional quaternionic matrices. In the early 2000s, the subject expanded into infinite-dimensional operator theory through the development of slice hyperholomorphic function theory. This led to the introduction of the S-spectrum and the quaternionic functional calculus by F. Colombo, I. Sabadini, providing a complete framework for the spectral theory of quaternionic linear operators. Recent work has also explored applications of this theory to fractional difussion processes.
Applications Quaternionic eigenvalue problems appear naturally in quaternionic quantum mechanics, where observables and evolution operators are represented by quaternionic linear operators. Quaternionic matrices are also used in applied mathematics and engineering, for instance colour image processing and signal processing.
Background
Quaternions
The quaternions, denoted by H {\displaystyle \mathbb {H} } , are a four-dimensional associative algebra over the real numbers. Every quaternion can be written uniquely in the form
q = a + b i + c j + d k , {\displaystyle q=a+bi+cj+dk,}
where a , b , c , d ∈ R {\displaystyle a,b,c,d\in \mathbb {R} } and the imaginary units satisfy
i 2 = j 2 = k 2 = i j k = − 1. {\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1.}
Unlike the real and complex numbers, quaternion multiplication is not commutative. For example,
i j = k , whereas j i = − k . {\displaystyle ij=k,\qquad {\text{whereas}}\qquad ji=-k.}
Every nonzero quaternion has a multiplicative inverse. The quaternions therefore form a division ring (also called skew field), but not a field. This makes quaternionic linear algebra distinct to real and complex linear algebra.
Quaternionic vector spaces Since quaternions form a noncommutative division ring, scalar multiplication on a quaternionic vector space must be specified to act either on the left or on the right. A left quaternionic vector space (or left H {\displaystyle \mathbb {H} } -module) is an abelian group equipped with a scalar multiplication
V × H → V ( v , q ) ↦ q v {\displaystyle {\begin{aligned}V\times \mathbb {H} &\to V\\(v,q)&\mapsto qv\end{aligned}}}
satisfying the vector space axioms. Similarly, a right quaternionic vector space (or right H {\displaystyle \mathbb {H} } -module) is an abelian group equipped with a scalar multiplication
V × H → V ( v , q ) ↦ v q {\displaystyle {\begin{aligned}V\times \mathbb {H} &\to V\\(v,q)&\mapsto vq\end{aligned}}}
satisfying the same axioms. Accordingly, two notions of quaternionic linear map arise. If V is a right quaternionic vector space, a map T : V → V {\displaystyle T:V\to V} is right-linear if
for every v , w ∈ V {\displaystyle v,w\in V} , T ( v + w ) = T ( v ) + T ( w ) , {\displaystyle T(v+w)=T(v)+T(w),}
for every v ∈ V {\displaystyle v\in V} and q ∈ H {\displaystyle q\in \mathbb {H} } , T ( v q ) = T ( v ) q . {\displaystyle T(vq)=T(v)q.}
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