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A is a square matrix with the following properties: 1. the diagonal elements are zero. 2. every element in the same row shares the same positive value.

What is the sufficient and necessary conditions for the convergence of the geometric series? the conditions, such as, the absolute value of eigenvalues is less than one, is not necessary.

Thanks.

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    Please state your question more carefully if you expect us to answer carefully. Where does your condition 2. end? And if the answer is "just before 'what'", then the condition is still not clear; can you give an example? Which geometric series are you talking about?2012-06-14
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    What does it mean for every element contained in the same row to share the same positive value? Also, $|\lambda_i|<1$ for each eigenvalue *is* both necessary and sufficient.2012-06-14
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    thanks, I mean every entry is the same except the diagonal one.2012-06-14
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    So matrices of the form $$\begin{pmatrix}r_1\\r_2\\ \vdots\\r_n\end{pmatrix}(1~~1~~\cdots~~1)-\begin{pmatrix}r_1 & 0 & \cdots & 0 & 0 \\ 0 & r_2 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & r_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & r_n\end{pmatrix},$$ if I understand you correctly.2012-06-14

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