Matrix pencils with symmetries: Nonstandard involution

This chapter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution φ‎. Here, the chapter provides canonical forms for φ‎-hermitian pencils, i.e., pencils of the form A + tB , where A and B are both φ‎-hermitian. It also provides canonical forms for φ‎-skewhermitian pencils. The canonical forms in question are with respect to either strict equivalence of pencils or to simultaneous φ‎-congruence of matrices. Applications are made to joint φ‎-numerical ranges of two φ‎-skewhermitian matrices and to the corresponding joint φ‎-numerical cones. The chapter fixes a nonstandard involution φ‎ throughout and a quaternion β‎(φ‎) such that φ‎=(β‎(φ‎)) = −β‎(φ‎) and ∣β‎(φ‎)∣ = 1.