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FM Transition matrices

2.3 Extended Matrix Recurrence Relations

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.

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2.2 Applications of Matrix Recurrence Relations

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

Transition Diagrams

• Transition diagrams provide a visual method for modelling how a system changes between states.
• They consist of a series of nodes (dots) representing the variables in the system, with arrows drawn between them showing the movement of individuals/objects in the system between variables (e.g. the percentage of people who go from playing soccer one month to playing baseball the next).
• By convention numerical values should be shown as percentages in transition diagrams.
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2.1 Simple Matrix Recurrence Relations

The Simple Matrix Recurrence Relation Formula

• The simplest type of matrix recurrence relation formula we will analyse in Further Maths models a system where the next “state”; S_{n+1}, can be reached by multiplying the current state; S_{n}, by a transition matrix; T, in the form:

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

• The state matrices; S_{n} (where n is a positive whole number representing the state of the system), are column matrix listing the value of each of the system’s variables in the corresponding state.
• The transition matrix; T, is a square matrix.
• As with a linear recurrence relation, it is important to state the initial state of a system; S_{0}.
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