Note: if you cannot remember how to use simple matrix recurrence relations, revise notes for 2.1 Simple Matrix Recurrence Relations.

### The Extended Matrix Recurrence Formula

- The
**simple matrix recurrence formula**we have analysed previously (in 2.1 Simple Matrix Recurrence Relations and 2.2 Applications of Matrix Recurrence Relations) only allows the values for the next state to be calculated based on proportional values of the current state. In some systems, this is not enough to produce an accurate model. The**extended matrix recurrence formula**allows us to model systems which have fixed values added or removed between states. - The
**general form**for the extended recurrence formula is:

S_{n+1}=T S_{n}+B

Where S_n is the state matrix for the nth state, T is the transition matrix and B is a matrix containing constant values.