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How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $?

The Frobenius inequality of linear algebra, with $A,B,C\in M_n(\mathbb{F})$, is: $$\operatorname{rank}{AB}+\operatorname{rank}{BC}\le\operatorname{rank}{B}+\operatorname{rank}{ABC}$$

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  • $\begingroup$ In wikipedia, I've already read the proof you referred to. What I don't understand is, what is $\ker{ABC}/\ker{BC}$ and why that projection is one-to-one? $\endgroup$
    – xzhu
    Sep 19, 2012 at 0:03
  • $\begingroup$ This is also a duplicate of math.stackexchange.com/questions/497830/… . $\endgroup$
    – darij grinberg
    Jul 23, 2016 at 15:40

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To answer your more focused question in the comment:

The clear candidate map is $x\mapsto Cx +\ker (B)$, which gives you a homomorphism from $\ker(ABC)\rightarrow \ker(AB)/\ker(B)$.

Modding out by this map's kernel, which happens to be $\ker(BC)$, the new map from $\ker(ABC)/\ker(BC)\rightarrow \ker(AB)/\ker(B)$ is necessarily an injection. (This is just an isomorphism theorem.)

I'll also add that there is no indication that it is (or even needs to be) onto, so it's not really a projection.

From that point on, presumably you are working with all finite dimensional spaces, so this isomorphism gives you the inequality $\dim(\ker(ABC))-\dim(\ker(BC))\leq \dim(\ker(AB))-\dim(\ker(B))$

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