Canonical Forms for Symmetric and Skewsymmetric Quaternionic Matrix Pencils

Comparison of congruences and strict equivalences for real, complex, and quaternionic matrix pencils with symmetries

“…This property, as well as and many other properties of iaa's to be used later on in the present paper, follows easily from the following known description of iaa's (see [15], [16], for example): In view of (1.3), indeed all nonstandard iaa's can be treated in one category (V). We also note that if σ is a nonstandard iaa of H, then there is a unique (up to multiplication by −1) quaternion β such that β 2 = −1 (this equaity holds if and only if β has norm 1 and zero real part) and σ(β) = −β.…”

“…For more information and proofs, we refer the reader to [22], [23], [16], [5], [15], among many other sources. Recent interest in quaternionic linear algebra is motivated in part by applications in system and control [13], [14].…”

Hamiltonian square roots of skew Hamiltonian quaternionic matrices

“…We denote by Inv (φ) the set of all quaternions fixed by φ; Inv (φ) is a three-dimensional real vector space spanned by 1, α 1 , α 2 , where α 1 , α 2 ∈ H are certain square roots of −1. (For these and other well-known properties of iaa's see, for example, [1], [16], or [17]. )…”

“…First, we describe the primitive forms: As with Proposition 3.1, several equivalent versions of the canonical form of Hskew-symmetric matrices are known; we mention here only the books [3], [4]; usually, they are derived from the canonical forms for pairs of quaternionic matrices, where one matrix is φ-symmetric and the other one is φ-skewsymmetric. A detailed proof of the canonical form as in Proposition 3.2 can be found in [18]. …”

Invariant neutral subspaces for Hamiltonian matrices

“…These forms, in various contexts and appearances, are well known for real and complex matrices, and are known (perhaps not well known) for quaternion matrices. Some references for these forms, by no means a complete list, are [2], [8], [9], [14], [15], [22], [23], [24], [28], and [29]. For the reader's convenience, the canonical forms needed are reproduced in the present paper.…”