# 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

Oct 24, 2017

${S}_{n} = {T}^{n} {S}_{0}$

#### Explanation:

We have from

${S}_{n} = T {S}_{n - 1}$
${S}_{n} = {T}^{k} {S}_{n - k}$
${S}_{n} = {T}^{n} {S}_{0}$

Oct 27, 2017

${S}_{n} = {T}^{n} {S}_{0}$

#### Explanation:

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

$A B C = A \left(B C\right) = \left(A B\right) C$

Therefore, the recurrence formula ${S}_{n} = T {S}_{n - 1}$ can be solved like that of a geometric progression.

In the recurrence formula for a geometric progression ${a}_{n} = r {a}_{n - 1}$, the general term is ${a}_{n} = {r}^{n} {a}_{0}$. Similarly, the general term of ${S}_{n}$ is ${S}_{n} = {T}^{n} {S}_{0}$.

i.e.
${S}_{1} = T {S}_{0}$
${S}_{2} = T {S}_{1} = T \left(T {S}_{0}\right) = \left(\text{TT}\right) {S}_{0} = {T}^{2} {S}_{0}$
${S}_{3} = T {S}_{2} = T \left({T}^{2} {S}_{0}\right) = {T}^{3} {S}_{0}$
and so on.

Note that $\textcolor{red}{\text{commutative law AB=BA cannot be applied for matrix multiplication.}}$ However, other rules(associative law, distributive law) can be used in multiplication.