Mixing problem and Eigen vectors relationship.

[dionysian]

Homework Statement

A mixing protocol for 5 containers of solvent consists of a repetition of the following procedure. The contents
of each container are removed and divided up into pre-defined fractions. The various fractions are then poured back
into the containers in a pre-assigned way. The idea is to continually mix up the solvents in the ve containers, such
that the concentration of added chemicals can be maintained at a constant level.
Don't worry if you did not catch all the details of the previous paragraph. Let us formulate the problem
mathematically, and hopefully that will clarify things for you. Let pn denote the row vector describing how much
solvent is in each container at the nth repetition of the mixing protocol. The rules for dividing up the contents and
then refilling them tell us that pn = Mpn-1, where

M = [0 0 2/7 1/2 1/9
1/5 0 1/7 1/4 1/9
2/5 1/4 1/7 1/4 1/3
0 1/2 2/7 0 1/3
2/5 1/4 1/7 0 1/9 ]

The matrix M is given to us by the person in charge of the mixing protocol. Assume that at the beginning all the
solvent is inside the rst container, that is, p1 = [1; 0; 0; 0; 0]. How can we determine the distribution of the solvent
after the first application of the mixing protocol? Well, we simply use our math, and compute p2 = Mp1, right?
Using matrix multiplication, we may determine p2, then p3, and so on. Now, do the following

(i) Compute the components of pn against n for the rst 15 mixes and draw a graphics showing them. You
probably will need to use a loop. What do you observe in the picture?
(ii) Compute the eigenvalues of M. Check that one of them is 1, whereas the other four have absolute values
that are less than one. The special eigenvalue 1 has an associated column eigenvector that solves the equation
v = Mv. Indicate how this eigenvector is related to the solution pn after many mixes, and hence deduce the
components of v.

Homework Equations

The Attempt at a Solution

I have done step one and got the proper paint distribution with the given instructions. MY ONLY QUESTION is this: I have no idea how the eigenvector associated with the eigenvalue 1 is related to the solution after many mixes.

To be honest i never really understood what the point of eigen vectors where... i have been told they are important and i have read about them online. This is my understanding of them. An eigen vector is the vector whos direction is not changeing under a transform.

Well, that is great and i see how the eigen vector in a rotation is the axis of rotation but there seems to be a more general idea behiend an eigen vector that eludes me. Can some one point my in the right direction on this? Possibly in the direction that would give me a solution to my problem?

 

[Dick]

If you take the initial vector and decompose it into eigenvectors the component with eigenvalue 1 will remain unchanged. The components with eigenvalues less than one will become less and less as the process is repeated.