# System of Equations

1. Nov 17, 2004

### LostInSpace

Hi! I'm trying to solve an equation system $$\vec{\pi}\mathbb{P} = \vec{\pi}$$ where $$\vec{\pi} = (\pi_1, \pi_2, \pi_3, \pi_4, \pi_5)$$ and $$\mathbb{P}$$ is a 5x5 matrix (constants). The problem is that the equation system is a bit to large to handle, at least for me. I remember that linear equation systems can be solved $$\mathbb{X}\vec{v} = \mathbb{Y} \Rightarrow \vec{v} = \mathbb{X}^{-1}\mathbb{Y}$$. Is there anything similiar I can use to solve this system? Or can I solve it using maple or matlab?

Nille

2. Nov 17, 2004

hi,
I am not sure whether i have correctly understood your problem.
I think u will be able to solve the system of equation iteratively.
consider,
Ax=b
where A is the constant 5 * 5 matrix, x is the variable vector(5 * 1) which you want to find out and b is again a constant vector(5 * 1).
The above equation can be written as,

(A+I - I)x = b, where I is the identity matrix.
simplifying we get,
x = (A+I)x -b.
Hence x(k+1) = (A+I)x(k) - b, where k is the iteration number.
One can start with some approximate value of the vector x at k=0.

If you are trying to solve something like:
Ax = x
this is equivalent to finding the eigen vector corresponding to eigenvalue 1 for the matrix A.