Primitive Matrix: Is M^k Positive?

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SUMMARY

The discussion centers on the matrix M, defined as the product of n matrices M_1, M_2, ..., M_n, each of dimension 2n by 2n. The matrix M_n is the identity matrix with a 3 by 3 block of positive elements in the bottom right corner. The inquiry focuses on whether M is primitive, specifically if there exists a k such that M^k is positive. MATLAB tests confirm that M is positive, leading to the conclusion that M is indeed a product of positive matrices, thus establishing that M is a positive matrix.

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bob j
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Hi All,
I have the following matrix

M = M_1 M_2 M_3 ... M_n

M is then a product of n matrices. Each of those has dimension 2n by 2n and has the same "look". Consider M_n: this matrix is equal to the identity matrix, 2n by 2n. The only thing different from the identity is that the 3 by 3 block on the bottom right of the matrix is composed of positive elements, and less than one. the other matrices are the same: matrix M_i has the 3 by 3 positive block placed at position 2(i-1) + 1, while the rest of the matrix is equal to the identity matrix.

My question is this. Is M primitive? Or, in other words, is M such that, for some k, M^k is positive? I tried with MATLAB and M is positive. Does anybody know if there is a theorem I can use?

Thanks
 
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If I'm reading your post correctly, it seems to me that M is a product of positive matrices, and hence is a positive matrix itself.
 

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