This paper studies necessary and sufficient conditions for
a matrix to be conjunctive with its adjoint. The problem is solved
completely in the usual complex case, in which it is shown that a
matrix is conjunctive to its adjoint iff it is conjunctive to a real
matrix. The problem is extended to pairs of fields , , where [: ] = 2 and characteristic ≠2. It is shown that
if a matrix is conjunctive to a matrix over is congruent over to its transpose. We
also show that it is sufficient to consider non-singular pencils by proving
the uniqueness up to conjunctivity of the non-singular summand of
the pencil λH + μK, where λ and μ are indeterminates over , H* = H and K*=K when λH + μK is decomposed (by
conjunctivity over ) into a direct sum of its minimum-indices
part and a non-singular part
