linear-algebraprobabilitymarkov-chainsmarkovmarkov-models

proof of each row of self product of transition matrix sums to 1


I am unable to proof that the sum of each row of self product of a transition matrix is 1...

Let A be a transition probability matrix which means that each row of A sums to 1, and let P=A*A.

I want the prove that P is a also a valid transition matrix,i.e each row of P sums to 1.

Please Help.

Regards.


Solution

  • Given two transition matrices A (m x p) and B (p x n), we want to prove that C = AB (m x n) is a transition matrix.

    We know that Cij = Σk Aik Bkj.

    What is the sum of the ith row of C?

    Σj Cij = Σj Σk Aik Bkj = Σk Σj Aik Bkj = Σk Aikj Bkj) = Σk Aik = 1

    Therefore C is a transition matrix.