Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I know that matrix multiplication is not commutative: $ A \cdot B \neq B \cdot A$

Still, it may be possible that $ \operatorname {rank} (A \cdot B) = \operatorname {rank} (B \cdot A) $ . But it seems to be not true.

How can it be proved that $ \operatorname {rank} (A \cdot B) \neq \operatorname {rank} (B \cdot A) $ ?

(Usually it's not a big deal to prove a negative statement since it only take to find one example when the equality is not satisfied)

share|improve this question
2  
Hint: you can do this with two-by-two matrices. Look for $A$, $B$ such that $AB=0$ but $BA \neq 0$. –  Chris Eagle Oct 31 '11 at 22:31
    
Try working with a single column full of $1$s and $0$s and a single row full of $1$s and $0$s. For ease, perhaps do the case where each is $2\times 1$ or $1\times 2$. –  Jason DeVito Oct 31 '11 at 22:32
    
Why do you think that this seems not to be true? To prove that something is not true in general, you need to find a single counterexample... –  N. S. Oct 31 '11 at 22:32
    
An other question is: determine $\max_{A\in\mathcal M_{n,p}(\mathbb R),B\in\mathcal M_{p,n}(\mathbb R)}\operatorname{rank}(A\cdot B)-\operatorname{rank}(B\cdot A)$ in terms of $n$ and $p$. –  Davide Giraudo Oct 31 '11 at 22:52
    
@JasonDeVito Thanks for the hit. I made an answer basing on it. Still, I want some intuition why it's so. We can look on the first matrix in a multiplication as on some operator. And why if A operates on B the rank is r1 and when B operates on A is r2, and r1 is not equal to r2. The proof is a good thing, but it'll be forgotten over time. And only intuition is what will be carried over time. –  ovgolovin Oct 31 '11 at 22:55
show 8 more comments

3 Answers

up vote 9 down vote accepted

Let $A=\pmatrix{0&1\\0&0}$ and let $B=\pmatrix{1&0\\0&0}$. These are matrices of simple geometric transformations.

If $v$ is a point in the plane, viewed as a vector, then $Bv$ is the projection of $v$ on the $x$-axis.

And if $w$ is a point in the plane, then $Aw$ is projection onto the $y$-axis, followed by reflection in the line $y=x$.

Now look at $A(Bv)$. We have (i) projected $v$ onto the $x$-axis, and then (ii) projected the result onto the $y$-axis, then reflected in the line $y=x$. Before the reflection, we are already at $(0,0)$, so $AB$ kills every vector.

Look next at $B(Av)$. If the $y$-component of $v$ is non-zero, $Av$ is a non-zero point on the $x$ axis, and the projection $B$ does nothing to $Av$. Thus $B(Av)$ is not the $0$-vector for "most" $v$.

share|improve this answer
add comment

To prove that $ \operatorname {rank} ( A \cdot B) \not= \operatorname {rank} ( B \cdot A) $ we just need to find one example when the equality is not satisfied.

As suggested in the comments to the answer, let's take $ A = \begin{pmatrix} 0 & 1\\ 0 & 1 \end{pmatrix} $ and $ B = \begin{pmatrix} 1 & 1\\ 0 & 0 \end{pmatrix} $.

$ A \cdot B = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} $ ; $ \operatorname {rank} ( A \cdot B) = 0 $ .

$ B \cdot A = \begin{pmatrix} 0 & 2\\ 0 & 0 \end{pmatrix} $ ; $ \operatorname {rank} ( B \cdot A) = 1 $ .

The ranks don't match, so $ \operatorname {rank} ( A \cdot B) \not= \operatorname {rank} ( B \cdot A) $ .

share|improve this answer
    
$A.B$ is surely [[1, 0], [0, 0]] –  Peter Taylor Oct 31 '11 at 23:09
add comment

You asked for "intuition". One way of looking at it is this.
An $m \times n$ matrix $A$ operates on $n$-dimensional vectors, taking them to $m$-dimensional vectors. The rank $r_A$ of $A$ is the dimension of the range of $A$, and is also $n$ minus the dimension of the kernel of $A$. That is, you start with $n$ linearly independent vectors $v_j$ spanning ${\mathbb R}^n$; the images $A v_j$ of them under $A$ are not linearly independent, because there are $k$ independent relations between them corresponding to the dimension of the kernel of $A$, so we have only $r_A = n - k$ linearly independent vectors spanning the range of $A$. Now in $BA$, $B$ operates on these vectors so the rank of $BA$ is the number of linearly independent vectors in the range of $BA$. This is reduced from $r_A$, not by the dimension of the kernel of $B$, but by the dimension of the intersection of the kernel of $B$ with the range of $A$. That is, $\text{rank}(BA) = \text{rank}(A) - \dim(\ker (B) \cap \text{range}(A))$. Similarly ($B$ being $n \times m$ so that $AB$ and $BA$ both make sense) $\text{rank}(AB) = \text{rank}(B) - \dim(\ker(A) \cap \text{range}(B))$. And there is no reason for these to be the same.

Indeed, given $n$ and the ranks of $A$ and $B$, $\ker(A)$ and $\text{range}(B)$ can be any subspaces of ${\mathbb R}^n$ of dimensions $n - \text{rank}(A)$ and $\text{rank}(B)$ respectively, and their intersection can have any dimension from $\max(\text{rank}(B) - \text{rank(A)},0)$ to $\min(n - \text{rank}(A),\text{rank}(B))$.

For example, if $m=n = 5$ and $\text{rank}(A) = \text{rank}(B) = 3$, $\dim(\ker(A) \cap \text{range}(B)$ can have any dimension from 0 to 2, making $\text{rank}(AB)$ anything from 1 to 3, and similarly $\text{rank}(BA)$ can be anything from 1 to 3. To get an example with $\text{rank}(AB) = 3$ and $\text{rank}(BA) = 1$, you want $\ker(A)$ and $\text{range}(B)$ to be subspaces of ${\mathbb R}^5$ of dimensions 2 and 3 respectively with intersection of dimension 0, while $\ker(B)$ and $\text{range}(A)$ are subspaces of dimensions 2 and 3 with intersection of dimension 2. If $e_1$ to $e_5$ are the standard basis vectors of ${\mathbb R}^5$, you could for example have $\ker(A) = \text{span}(e_1,e_2)$ while $\ker(B) = \text{span}(e_4,e_5)$ with $\text{range}(A) = \text{range}(B) = \text{span}(e_3, e_4, e_5)$. Thus we could have $$ A= \left[ \begin {array}{ccccc} 0&0&0&0&0\\ 0&0&0&0&0 \\ 0&0&1&0&0\\ 0&0&0&1&0 \\ 0&0&0&0&1\end {array} \right], \ B = \left[ \begin {array}{ccccc} 0&0&0&0&0\\ 0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0 \\ 0&0&1&0&0\end {array} \right] $$

share|improve this answer
    
Thank you for the answer. I even started watching videos on the Khan Academy site about the nullspace of a matrix after reading you answer! After watching this video I'll recur to your answer and hope I'll managed to grasp it completely. –  ovgolovin Nov 1 '11 at 23:26
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.