The Sylvester Rank Inequality
We prove the Sylvester Rank Inequality. This asserts that if A is an mxk matrix and B is a kxn matrix, then rank(A) + rank(B) is no more than k + rank(AB). This Inequality is proven using the Rank-Nullity Theorem along with an inequality for nullity(AB) in terms nullity(A) and nullity(B). This result gives us good information about the range of factorizations for non-square matrices.
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