Primitive Matrix: Is M^k Positive?

[bob j]

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

 

[morphism]

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.