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I have been attempting to prove Sylvester's Inequality. $$rank(A)+rank(B)\leq n+rank(AB)$$

I have been referencing other proofs on this site, but all of them involve concepts and theorems that I am not familiar with, so I am still struggling.

So far, I have proven that $rank(AB) \le rank(A)$ and $rank(AB) \le rank(B)$. I also have rewritten the inequality as $ker(A)+ker(B) \ge ker(AB)$, but I can't seem to get past this point. Any help would be appreciated.

Skipher
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  • You have proofs; they require concepts and theorems you don't know. One (quite good) solution is to learn those concepts and get familiar with those theorems. Anything we'd write here would probably amount to a rewrite of their key points anyhow. Unless you tell me that there's some urgent reason for you to understand a proof immediately, it's hard to get motivated to do that work. – John Hughes Sep 02 '17 at 03:29
  • I did not mean to say that I am too lazy to learn the theorems and concepts myself, but I need help understanding those as well. Taking the 2nd answer for this question for example, https://math.stackexchange.com/questions/269474/prove-that-textrankab-ge-textranka-textrankb-n , I understand that $\beta$ is the basis of ker(B), but I am at a loss beginning with the basis of ker(AB). Would you be kind as to provide a simpler explanation for this proof? Thank you. In a sense, my question arose in response to my inability to understand other answers. – Skipher Sep 02 '17 at 03:34

1 Answers1

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You can avoid using "advanced" theorems if you do the following steps:

Let $A$ be $m\times n$ matrix and $B$ be $n\times k$ matrix.

  1. Prove that $$ \operatorname{rank}\begin{bmatrix}I_n & 0\\0 & AB\end{bmatrix}=n+\operatorname{rank}(AB). $$
  2. Prove that $$ \operatorname{rank}\begin{bmatrix}I_n & 0\\0 & AB\end{bmatrix}=\operatorname{rank}\begin{bmatrix}I_n & B\\A & 0\end{bmatrix} $$ using e.g. $$ \begin{bmatrix}I_n & B\\A & 0\end{bmatrix}=\begin{bmatrix}I_n & 0\\A & I_m\end{bmatrix}\begin{bmatrix}I_n & 0\\0 & AB\end{bmatrix}\begin{bmatrix}I_n & B\\0 & -I_k\end{bmatrix}. $$
  3. Prove that $$ \operatorname{rank}\begin{bmatrix}I_n & B\\A & 0\end{bmatrix}\ge \operatorname{rank}(A)+\operatorname{rank}(B). $$

The last step can be done easily if you prove that the columns of the block matrix corresponding to the linear independent columns of $A$ and that of $B$ are linear independent altogether. All you need here is the definition of linear independent vectors.

A.Γ.
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