Minimal Polynomials of Some Matrices Via Quaternions

TL;DR

This paper characterizes minimal polynomials of various structured 4x4 matrices using quaternion algebra, providing explicit formulas and uncovering new matrix classes with similar polynomial behavior.

Contribution

It introduces explicit minimal polynomial formulas for structured 4x4 matrices via quaternion algebra and identifies new matrix classes with similar polynomial properties.

Findings

Explicit formulas for minimal polynomials of symmetric, Hamiltonian, and orthogonal matrices.

Complete determination of Jordan structure for skew-Hamiltonian matrices.

Identification of new matrix classes with minimal polynomial behavior akin to known classes.

Abstract

This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete determination of the Jordan structure of skew-Hamiltonian matrices and the computation of the Cayley transform are given. Some new classes of matrices are uncovered, whose behaviour insofar as minimal polynomials are concerned, is remarkably similar to those of skew-Hamiltonian and Hamiltonian matrices. The main technique is the invocation of the associative algebra isomorphism between the tensor product of the quaternions with themselves and the algebra of real $4\times 4$ matrices.