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$A$ and $B$ are two stochastic non-negative matrices. $B$ has the same value as $A$ but some rows and columns are displaced.

I was wondering if there is a way to proof: $\mathrm{SLEM}(A \cdot B) < \mathrm{SLEM}(A \cdot A)$

$\mathrm{SLEM}$ is second largest eigenvalue

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    Do you know that saying that $B$ results from $A$ by moving some rows and columns is equivalent to say that $B=P_1AP_2$ where $P_1$ and $P_2$ are certain permutation matrices (https://en.wikipedia.org/wiki/Permutation_matrix)? – 2017-02-08
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    @JeanMarie: What is it the connection of the eigenvalues of $AP_1AP_2$ and $AA$? – 2017-02-08
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    I haven't said I know a connection. I just indicated a standard way to put in matricial form the fact that you move rows ans columns. – 2017-02-09
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    as an example matrix A is : – 2017-02-09
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    as an example matrix A is : [1/3 1/3 1/3 0 0 0 0;1/3 1/3 1/3 0 0 0 0; 1/6 1/6 4/15 1/10 1/10 1/10 1/10 ; 0 0 1/5 1/5 1/5 1/5 1/5; 0 0 1/5 1/5 1/5 1/5 1/5;0 0 1/5 1/5 1/5 1/5 1/5;0 0 1/5 1/5 1/5 1/5 1/5;] – 2017-02-09
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    an example of matrix B is – 2017-02-09
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    1/5 0 1/5 0 1/5 1/5 1/5 – 2017-02-09
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    0 1/3 1/3 1/3 0 0 0 – 2017-02-09
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    1/10 1/6 4/15 1/6 1/10 1/10 1/10 – 2017-02-09
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    0 1/3 1/3 1/3 0 0 0 – 2017-02-09
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    1/5 0 1/5 0 1/5 1/5 1/5 – 2017-02-09
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    1/5 0 1/5 0 1/5 1/5 1/5 – 2017-02-09
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    1/5 0 1/5 0 1/5 1/5 1/5 – 2017-02-09
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    As you can see, the values didn't change and the change is only in their location. – 2017-02-09

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