# Math Help - finding a steady state matrix

1. ## finding a steady state matrix

A security guard is employed to patrol a shopping complex. The guard
is instructed to wait 10 minutes at each corner (numbered 1 to 4.)
After 10 minutes, the guard must either stay where he is or move to
one of the adjacent corners. Movements should be at random so that the
chances of remaining or moving to each adjoining corner are the same.

What is the steady-state matrix for the likelihood of the guard's being
at each of the corners?

Other information given:

the steady state vector is defined as s = lim (n -> infinity) of x^n.
Since s is independent from initial conditions, it must be unchanged when
transformed by P i.e sP = s.

If someone could briefly explain this as well that would be great.

2. Hello,
Originally Posted by Gul
A security guard is employed to patrol a shopping complex. The guard
is instructed to wait 10 minutes at each corner (numbered 1 to 4.)
After 10 minutes, the guard must either stay where he is or move to
one of the adjacent corners. Movements should be at random so that the
chances of remaining or moving to each adjoining corner are the same.

What is the steady-state matrix for the likelihood of the guard's being
at each of the corners?

Other information given:

the steady state vector is defined as s = lim (n -> infinity) of x^n.
Since s is independent from initial conditions, it must be unchanged when
transformed by P i.e sP = s.

If someone could briefly explain this as well that would be great.
I don't quite understand what x is...

Anyway, let's find the transition matrix P...
The adjacent corners to 1 are 4 and 2
The ac to 2 are 1 & 3
The ac to 3 are 2 & 4
The ac to 4 are 3 & 1

So we have $P=\begin{pmatrix} 1/3 & 1/3 & 0 & 1/3 \\ 1/3&1/3&1/3&0 \\ 0&1/3&1/3&1/3 \\ 1/3&0&1/3&1/3 \end{pmatrix}$

Just find $(a,b,c,d)$ such that $(a,b,c,d)P=(a,b,c,d)$

This is just a simple matrix multiplication.

If I misunderstood what you don't understand, just tell me, I'll try to help as much as possible...