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What kind of properties does a matrix whose column entries sum to zero have?

$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}$$

Where $a_{11}+\cdots+a_{m1}=0$ and so on.

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You have $$a_{mk} = - \sum_{j=1}^{m-1} a_{jk}$$ for all $k$. This means the last row vector is a linear combination of the remaining row vectors. Hence, the rank of the matrix is at most $\min(m,n)-1$.

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  • $\begingroup$ What!? How do you get that? $\endgroup$ – Yosef Qian May 18 '13 at 0:01
  • $\begingroup$ @user17762 Don't you mean $m-1$? $\endgroup$ – Trancot May 22 '13 at 19:12
  • $\begingroup$ @Trancot Nope, it is $\min(m,n)-1$. For instance, consider the matrix $$\begin{bmatrix}1 & -1\\ 2 & -2\\ 3 & -3\\ 4 & -4 \end{bmatrix}$$ Here $m=4$ and $n=2$. But rank is $n-1=1$. $\endgroup$ – user17762 May 22 '13 at 19:58
  • $\begingroup$ @user17762, Oh, no, I mean the upper index on your sum, namely $\overset{\overset{?}{n-1}}{\sum}$. $\endgroup$ – Trancot May 22 '13 at 20:26
  • $\begingroup$ @Trancot Oh yes. Thanks. Corrected. $\endgroup$ – user17762 May 22 '13 at 20:34

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