Minimal Polynomials of Some Matrices Via Quaternions

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.