# Destroying the structure of a linear system while preserving its maximum eigenvalue

I have an asymmetric square matrix with non-negative real entries in the range [0,10], representing the edge-weights of a directed network. Assume that the network is a linear system. My general question is the following: how far can I go about destroying any structural patterns present in the network, while preserving its maximum eigenvalue?

Say I have an asymmetric 10 X 10 matrix M, whose absolute value of the maximum eigenvalue is v. Say M has 70 non-zero entries in the range (0,10] and the rest 0. Now, the questions:

(1) Does v depend on the number of non-zero entries of the matrix? So suppose I randomly replace some n zero-entries with values from (0,10], or replace n non-zero entries with 0, does the maximum eigenvalue v of the new matrix change predictably with n?

(2) Does v depend on the sum of non-zero entries of M? So if I randomly pick pairs of non-zero entries, and replace one of them with their sum and the other with 0, will v be sensitive to it?

(3) Does v depend on the spatial distribution or configuration of the non-zero entries of M? So, if I randomly shuffle around the non-zero entries, how much can I expect v to vary? Are there any bounds to v subject to all possible shuffling configurations?

If the answers are NO to the above questions, then I will have at least 3 ways of modifying (possibly randomly) the connection structure and the edge weights of M, while subjecting it to the constraint that its maximum eigenvalue v stays constant or at least within some range v +/- x.

Danke!

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Seems to me that all of your proposed modifications can lead to big changes in the maximum absolute value of the eigenvalues. For (1), consider $$\pmatrix{10&0\cr0&1\cr}{\rm\ and\ }\pmatrix{0&0\cr0&1\cr}$$ For (2), $$\pmatrix{5&0\cr0&5\cr}{\rm\ and\ }\pmatrix{10&0\cr0&0\cr}$$ For (3), $$\pmatrix{10&1\cr0&1\cr}{\rm\ and\ }\pmatrix{1&10\cr0&1\cr}$$