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.
Best Answer
That all eigenvalues $\lambda_i$ of a matrix $A$ are strictly $<1$ in abs. value is both necessary and sufficient.
A couple of items require some further justification using the underlying topology (defined e.g. by the metric induced by the Frobenius norm): continuity of matrix multiplication (in the example this should be clear because matrix multiplication is a linear map..), and the fact that eigenvalues approaching zero implies a sequence of matrices converges to zero (perhaps we could consider the action of $S_n$ on the subspaces in the Jordan aka generalized eigenvector decomposition?).
(There are probably some standard arguments for these things, but I may not have hit them all because I'm pulling them out of my hat instead of going on experience here.)