Three New Proofs of the Theorem rank f ( M ) + rank g ( M ) = rank ( f, g )( M ) + rank [ f, g ]( M )

Vasile Pop and Alexandru Negrescu ()
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Vasile Pop: Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Alexandru Negrescu: Department of Mathematical Methods and Models, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania

Mathematics, 2024, vol. 12, issue 3, 1-6

Abstract: It is well known that in C [ X ] , the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications.

Keywords: rank; Jordan canonical form; matrix polynomials; Sylvester inequality; Frobenius inequality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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