Problem 1179

Is the following transition matrix regular?

A = \begin{bmatrix}  1 & 0  \\  0.75& 0.25  \end{bmatrix}




A transition matrix is regular if same power of the matrix contains all positive entries.

The given matrix is A, or A1. Since 0 is not a positive number, A1 does not contain all positive entries.

Thus, other powers of a must be examined to determine whether or not A is regular.

Next, look at A2, which if found by multiplying A by itself using matrix multiplication.

A2 = \begin{bmatrix}  1 & 0  \\  0.9375& 0.0625  \end{bmatrix}


Notice that both A and A2 contain only one zero and in both matrices this zero is in row 1 and column 2. For any transition matrix P, if all zeros occur in the identical places in both Pn and Pn+1 for any n, they will appear in those places for all higher powers of P. Therefore there would be no power of P that contain all positive entries, so P is  not regular.

Thus, the given transition matrix A is nor regular because the only 0 occurs in the identical place in both A1 and A2, which means there is no power of A that contains all positive entries.

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