I need to merge two states in the Transition Matrix:
For example: I have the matrix below
A B C D E F
A 0.5 0.4 0 0 0.1 0
B 0.5 0.1 0.2 0.1 0.1 0
C 0 0.1 0.9 0 0 0
D 0 0 0 0.7 0.3 0
E 0 0.2 0 0.7 0 0.1
F 0 0 0 0.5 0 0.5
And I want to join the states $D$ and $E$:
A B C (D+E) F
A 0.5 0.4 0 ? 0
B 0.5 0.1 0.2 ? 0
C 0 0.1 0.9 ? 0
(D+E) ? ? ? ? ?
F 0 0 0 ? 0.5
What are the formulas to obtain the row and column $(D+E)$?
Using the constraint: "the sum over column must be equal to $1$" is simple to calculate the elements: $$ (A,(D+E))=0.2 \\ (B,(D+E))=0.2\\ (C,(D+E))=0.1\\ (F,(D+E))=0.5\\ $$ How can I calculate the elements of row $((D+E),i)$?