Matrices conjunctive with their adjoints Public Deposited

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  • 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 𝓕, 𝓔, where [𝓕: 𝓔] = 2 and characteristic 𝓕≠2. It is shown that if a matrix is conjunctive to a matrix over 𝓔 is congruent over 𝓔 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 𝓔, H* = H and K*=K when λH + μK is decomposed (by conjunctivity over 𝓕) into a direct sum of its minimum-indices part and a non-singular part.
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