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If we have a matrix A of $n\times m$, (with n number of rows and m number columns of the matrix A) and $n

If, in the matrix A, we have many submatrix of $i\times i$, $i

For instance $$A=\begin{pmatrix}0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 &0 &\ldots\\ 0& 0& 0& 0& a_1 & a_2 & a_3 & a_{6} & a_{7} & a_{8} \\0 & a_1 & a_2 & a_3 & 0 & 0 & 0 & a_{10} & a_{11} & a_{12}\\ a_1 & 0 & a_6 & a_7 & 0 & a_{10} & a_{11} & 0 & 0 & a_{13} \\a_2 & a_6 & 0 & a_8 & a_{10} & 0 & a_{12} & 0 & a_{13} & 0\\a_3 & a_7 & a_8 & 0 & a_{11} & a_{12} & 0 & a_{13} & 0 & 0 & \ldots\end{pmatrix}$$

Where $a_i\neq 0$. We have three of such submatrix, namely. $$\begin{pmatrix}0 & a_1 & a_2 & a_3\\a_1 & 0 & a_6 & a_7\\a_2 & a_6 & 0 &a_8\\a_3 & a_7 & a_8 & 0\end{pmatrix}, \begin{pmatrix}0 & a_{10} & a_{11}\\a_{10} & 0 & a_{12}\\a_{11} & a_{12} & 0\end{pmatrix}, \begin{pmatrix}0 & a_{13}\\a_{13} & 0\end{pmatrix}$$

How I can find a lower bound R, (different of zero) for the rank of A, i.e. $R\leq rank[A]$. Any idea will be aprecicated. Thanks!

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    Where is the $i \times i$ submatrix located in your example? – 2017-02-19
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    There are three diferent submatrix. One of $4\times 4$, $3\times 3$ and $2\times 2$. For instance , $\begin{pmatrix} 0& a_1& a_2 & a_3\\a_1 & 0 & a_6 & a_7\\a_2 & a_6 & 0&a_8\\a_3 & a_7 & a_8 & 0\end{pmatrix}$ – 2017-02-19
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    In the general statement, you're only assuming that there is one such submatrix, right? – 2017-02-19
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    No, in the principal matrix I have many of such submatrix. – 2017-02-19
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    I don't see how this can be turned into a precise question, then. You'd need to better define your hypotheses. – 2017-02-19
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    I think you are using the symbol $n$ with two diferent meanings, hess. Please edit to disambiguate. – 2017-02-19

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The only lower bound you can give, in general, is zero, since, after all, all of the entries could be zero. Even if your $a_i$ are all non-zero, the submatrices could be singular, e.g., $$\pmatrix{0&1&1&1\cr1&0&9&4\cr1&9&0&1\cr1&4&1&0\cr}$$ has rank three (if I've done my arithmetic correctly). So even if your $a_i$ are nonzero, and your submatrices are $4\times4$, the rank could be as small as three.