Mathematics, Vol. 12, Pages 360: Three New Proofs of the Theorem rank f(M) + rank g(M) = rank (f,g)(M) + rank [f,g](M)

Mathematics, Vol. 12, Pages 360: Three New Proofs of the Theorem rank f(M) + rank g(M) = rank (f,g)(M) + rank [f,g](M)

Mathematics doi: 10.3390/math12030360

Authors: Vasile Pop Alexandru Negrescu

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.