Lumpability - Example

Example

Consider the matrix

P = \begin{pmatrix} \frac{1}{2} & \frac{3}{8} & \frac{1}{16} & \frac{1}{16} \\ \frac{7}{16} & \frac{7}{16} & 0 & \frac{1}{8} \\ \frac{1}{16} & 0 & \frac{1}{2} & \frac{7}{16} \\ 0 & \frac{1}{16} & \frac{3}{8} & \frac{9}{16} \end{pmatrix}

and notice it is lumpable on the partition t = {(1,2),(3,4)} so we write

P_t = \begin{pmatrix} \frac{7}{8} & \frac{1}{8} \\ \frac{1}{16} & \frac{15}{16} \end{pmatrix}

and call Pt the lumped matrix of P on t.

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