If A and B are two matrices of the same order n, then
rankA+rankB≤rankAB+n.
I don't know how to start proving this inequality. I would be very pleased if someone helps me. Thanks!
Edit I. Rank of A is the same of the equivalent matrix A′=(Ir000). Analogously for B, ranks of A and B are r,s≤n. Hence, since rankAB=min, then r+s\leq \min\{r,s\} + n. (This is not correct since \operatorname{rank} AB \leq \min\{r,s\}.
Edit II. A discussion on the rank of a product of H_f(A) and H_c(B) would correct this, but I don't know how to formalize that \operatorname{rank}H_f(A) +\operatorname{rank}H_c(B) - n \leq \operatorname{rank}[H_f(A)H_c(B)].
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