Powers of Transition Matrices (DP IB Applications & Interpretation (AI)): Revision Note

Powers of Transition Matrices

How do I find powers of a transition matrix?

• You can simply use your GDC to find given powers of a matrix

• The power could be left in terms of an unknown n

  • In this case it would be more helpful to write the transition matrix in diagonalised form (see section 1.8.2 Applications of Matrices) T = PDP^-1 where

    • D is a diagonal matrix of the eigenvalues

    • P is a matrix of corresponding eigenvectors

  • Then Tⁿ = PDⁿP^-1

    • This is given in the formula booklet

  • Every transition matrix always has an eigenvalue equal to 1

What is represented by the powers of a transition matrix?

• The powers of a transition matrix also represent probabilities

• The element of Tⁿ in the i^th row and j^th column gives the probability tⁿ_ij of :

  • the future state after n intervals of time being the state corresponding to row i

  • given that the current state is the state corresponding to column j

• For example: Let T be a transition matrix with the element t_2,3 representing the probability that tomorrow is sunny given that it is raining today

  • The element t⁵_2,3 of the matrix T⁵ represents the probability that it is sunny in 5 days’ time given that it is raining today

• The probabilities in each column must still add up to 1

How do I find the column state matrices?

• The column state matrix s_n is a column vector which contains the probabilities of each state being chosen after n intervals of time given the current state

  • s_n depends on s₀

• To calculate the column state matrix you raise the transition matrix to the power n and multiply by the initial state matrix

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    • You are given this in the formula booklet

• You can multiply s_n by the fixed population size to find the expected number of members of the population at each state after n intervals of time