How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$

I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ interpreting $AB$ as a composition of linear maps, observing that $\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.

How to manage $\operatorname{rank}(AB) \leq \operatorname{rank}(A)$? Is there a nice interpretation like the previous one?

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Your proof is fine. Furthermore, the same reasoning will get your desired fact. Again rank-nullity will tell you that the dimension of your vector space minus the dimension of the kernel will give you the rank. – BBischof Jul 28 '10 at 16:08

Yes. If you think of A and B as linear maps, then the domain of A is certainly at least as big as the image of B. Thus when we apply A to either of these things, we should get "more stuff" in the former case, as the former is bigger than the latter.

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Thank you. I was so obsessed with the kernel-image dimension formula that I could't recognize this simple fact. – user365 Jul 30 '10 at 8:52

Prove first that if $f:X\to Y$ and $g:Y\to Z$ are functions between finite sets, then $|g(f(X))| \leq \min \{ |f(X)|, |g(Y)| \}.$

Then use the same idea.

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Categorification... :-) – Kevin H. Lin Jul 29 '10 at 21:50
I am not familiar with the 'categorification'. How can one go from this to the rank inequality ? What functor is to be applied ? – nicolas Apr 13 '14 at 12:10
So you're saying that rank of linear map is somehow like image of a function? – Mihail Feb 12 '15 at 10:15
@Mihail, it is like the size of the image of a function. in fact, if $f$ is a linear map, the rank of $f$ is the dimension of the image of $f$ and the dimension is a measure of the size of a space. – Mariano Suárez-Alvarez Feb 12 '15 at 18:22

Once you have proved $\operatorname{rank}(AB) \le \operatorname{rank}(B)$, you can obtain the other inequality by using transposition and the fact that it doesn't change the rank (see e.g. this question).

Specifically, letting $C=A^T$ and $D=B^T$, we have that $\operatorname{rank}(DC) \le \operatorname{rank}(C) \implies \operatorname{rank}(C^TD^T)\le \operatorname{rank} (C^T)$, which is $\operatorname{rank}(AB) \le \operatorname{rank}(A)$.

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Very nice! Thank you. – user365 Sep 2 '10 at 10:01
I am doing this problem now too. Why is proving both inequalities enough? Where does the proof take the min idea into account? – Ryles2014 Apr 12 at 3:27

Let $m \le n, A \in M_{m\times n}, B\in M_{n\times m}$.

$\mbox{rank } A\le m$ and $\text{rank }B\le m$. (Obvious fact as rank A = dimension of the columnspace of A = dimension of the row space of A.)

Let $E_{n\times n}B$ be the row echelon form of $B$ and let $AE_{m\times m}$ be the column echelon form of $A$. ($E_{n\times n} ,E_{m\times m}$ are elementary matrices.)

We know $\operatorname{rank}(BA)=\operatorname{rank}(E_{n\times n}BA )=\operatorname{rank}(E_{n\times n}BAE_{m\times m} )$.

But $E_{n\times n}BAE_{m\times m} =\begin{pmatrix} L&0\\ 0&0\\ \end{pmatrix},$ where $L$ is an $k\times l$ matrix with $k\le \operatorname{rank}(B),l\le \operatorname{rank}(A)$.

So $\operatorname{rank}(E_{n\times n}BAE_{m\times m} )=\operatorname{rank}\begin{pmatrix} L&0\\ 0&0\\ \end{pmatrix}\le \min\{k,l\}\le \min\{\mbox{rank } A,\mbox{rank }B\}.$

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Here is another simple answer. When you multiply a matrix and a vector $Ax$ you end up with a linear combination of the columns of $A$.

$$Ax = \; x_1\,A_1 \;+\; x_2\,A_2 \;+\; x_3\,A_3 \;+\;\; ...\;\; \\$$

When we multiply two matrices $AB = C$, we have $AB_i = C_i$, which means that each column of $C$ is a linear combination of the columns of $A$, so $\text{rank}(AB) \leq \text{rank}(A)$. To show that $\text{rank}(AB) \leq \text{rank}(B)$ we follow a similar argument -- when you multiply $x^{\top}B$, you end up with a linear combination of the rows of $B$.

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Note that Col(AB) ⊆ Col(A) since given y ∈ Col(AB) we can choose x ∈ F and then we have y = (AB)x = A(Bx) ∈ Col(A).

Since Col(AB) ⊆ Col(A), any basis for Col(AB) can be extended to a basis for Col(A) and so dim Col(AB) ≤ dim Col(A), that is rank(AB) ≤ rank(A).

Note that Null(B) ⊆ Null(AB) since given x ∈ Null(B) we have (AB)x = A(Bx) = A 0 = 0 so that x ∈ Null(AB).

Since Null(B) ⊆ Null(AB), as above we have dim Null(B) ≤ dim Null(AB), that is, nullity(B) ≤ nullity(AB). Thus rank(AB) = n − nullity(AB) ≤ n − nullity(B) = rank(B).

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You should use MathJax to format your answer. – Michael Albanese Oct 15 '15 at 21:06

Already you have proved $$rank(AB)\leq rank(B)$$

For other part

rank of $A=$ dim range $A$

As range $AB \subset$ range $A\implies$ dim range $AB\leq$ dim range $A$ . Hence $$rank(AB)\leq rank (A)$$

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protected by Alexander Gruber♦Jul 5 '13 at 16:16

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