Given a strictly diagonally dominant matrix $A$ i.e., $|a_{ii}| > \sum_{j=1,i \neq j}^{n} |a_{ij}|$ for $i = 1 \cdots n$. I need to show that there is atleast one column k which is dominant i.e to show $|a_{kk}| > \sum_{j=1,k \neq j}^{n} |a_{jk}|$ for at least one value of $k = 1 \cdots n$
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Each diagonal entry is greater than the sum of the remaining entries in its row. Thus the sum of the diagonal entries is greater than the sum of all remaining entries. If each diagonal entry were less than or equal to the sum of the remaining entries in its row, the sum of the diagonal entries would have to be less than or equal to the sum of the remaining entries. (I omitted the absolute values for clarity.) |
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