I am confused about a transition matrix as I need to prove that if matrix A is positive, then A^(m+1) is also positive. However, when calculating the (m+1)th transition, I need to put matrix A on the left side of equation (A^m)x=x to write A(A^m)x=x. This to me represents after m times transitions, we are transitioning for the (m+1) times. Intuitively, this yields the result A^(m+1) because we essentially transitioned m+1 times following this matrix. However, mathematically, I can only say that A(A^m) = A^(1+m), which seems to be different from A^(m+1) because of the none-commutative property. Yet they seem to represent the same thing? Are they really equal in this case?