On 2-dimensional invariant subspaces of matrices
Omar Al-Raisi, Mohammad Shahryari
TL;DR
The paper addresses the problem of understanding 2-dimensional invariant subspaces (super-eigenvectors) of matrices over a non-commutative ring by developing a free $R$-module framework with $R=\mathrm{Mat}_2(\mathbb{F})$. It shows a correspondence between eigenstructures of matrices with entries in $R$ and 2D invariant subspaces of matrices over $\mathbb{F}$ via an $R$-basis and a mapping to $n\times n$ matrices over $\mathbb{F}$, enabling a determinant-based criterion using a quadratic polynomial $p_\Lambda(T)=T^2-\operatorname{tr}(\Lambda)T+\det(\Lambda)$. A key contribution is the criterion that, in $\operatorname{char}(\mathbb{F})\neq 2$, a proper super-eigenvalue $\Lambda$ exists iff $p_\Lambda(A)$ is singular and $p_\Lambda(T)$ is irreducible, tying proper 2D eigenstructure to irreducible quadratic factors of $p_A(T)$. The paper also provides a constructive finite bound on the number of non-similar proper super-eigenvalues and demonstrates the theory with a real example that yields two non-similar proper super-eigenvalues and explicit 2D invariant subspaces, highlighting the practical computability of the framework.
Abstract
We introduce a unified method for study of 2-dimensional invariant subspaces of matrices and their corresponding super-eigenvalues. As a novel application to non-commutative algebra, we present a connection between the eigenvalues of matrices with entries in the ring Mat_2(F) and 2-dimensional invariant subspaces of matrices with entries in the field F.