A transition matrix T tells us how to get from one state to another. That is Sn=TSn-1, where Sn is the distribution vector at time n and Sn-1 is the distribution vector at time n-1. We can conclude? Sn=TSn/2 Sn=TS0 Sn=TnSn-1 Sn=TnS0

A transition matrix T tells us how to get from one state to another. That is Sn=TSn-1, where Sn is the distribution vector at time n and Sn-1 is the distribution vector at time n-1. We can conclude?
Sn=TSn/2
Sn=TS0
Sn=TnSn-1
Sn=TnS0

2 Answers
Oct 24, 2017

#S_n = T^n S_0#

Explanation:

We have from

#S_n = T S_(n-1) #
#S_n= T^kS_(n-k) #
#S_n= T^n S_0#

Oct 27, 2017

#S_n=T^nS_0#

Explanation:

In the calculation of matrix, associative law can be applied as if it were a number.

#ABC=A(BC)=(AB)C#

Therefore, the recurrence formula #S_n=TS_(n-1)# can be solved like that of a geometric progression.

In the recurrence formula for a geometric progression #a_n=ra_(n-1)#, the general term is #a_n=r^na_0#. Similarly, the general term of #S_n# is #S_n=T^nS_0#.

i.e.
#S_1=TS_0#
#S_2=TS_1=T(TS_0)=("TT")S_0=T^2S_0#
#S_3=TS_2=T(T^2S_0)=T^3S_0#
and so on.

See also:
http://stattrek.com/matrix-algebra/matrix-theorems.aspx

Note that #color(red)"commutative law AB=BA cannot be applied for matrix multiplication."# However, other rules(associative law, distributive law) can be used in multiplication.