Abstract
In combinatorial studies of the Kronecker form of matrix pencils, two linear-algebraic characteristics have been featured: degrees of subdeterminants and ranks of expanded matrices. This paper shows the discrete Legendre duality between the two and their combinatorial counterparts for matroid pencils, which serve as upper bounds on the corresponding linear-algebraic quantities. Tightness of one of the combinatorial bounds is shown to be equivalent to that of the other. A sufficient condition for the tightness is given, and its application to electric networks is indicated. Furthermore, the proposed approach is extended to mixed matrix pencils.
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