Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence Original Research Article

A systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals, one of which is skew-symmetric and the other is either symmetric or skew-symmetric. These reductions are by strict equivalence and by congruence, over the reals or over the complex numbers, and essentially complete proofs are presented. The proofs are based on canonical forms attributed to Jordan and Kronecker. Some closely related results which can be derived from the canonical forms of pairs of symmetric/skew-symmetric real forms are also included. They concern simultaneously neutral subspaces, Hamiltonian and skew-Hamiltonian matrices, and canonical structures of real matrices which are selfadjoint or skew-adjoint in a regular skew-symmetric indefinite inner product, and real matrices which are skew-adjoint in a regular symmetric indefinite inner product. The paper is largely expository, and continues the comprehensive account of the reduction of pairs of matrices started in [P. Lancaster, L. Rodman, Canonical forms for hermitian matrix pairs under strict equivalence and congruence, SIAM Rev., in press].