#formularize

Ok. I'm taking a linear algebra class, just left class today, and my professor is a little (or a lot) kooky. He uses the Markhov matrices to find populations. Quite orthodox, of course. But the populations are Arachnids who are taking over the world in about 1000 years. Anyways, I usually feel the need to formularize my problem-solving approach (SI leader probs?). Here we go. This is my method for finding out how many Arachnids (adults and babies) exist in say, 1000 years. Can be used for other populations.

1. Set up the matrix, A. Make sure the columns add up 1. (There are exceptions; I am in an undergraduate class and I haven't been introduced to them). They will be in decimals or fractions, but convert the fractions to decimals, please.

2. Find the eigenvalues (λ - lambda) of A.

3. Use the eigenvalues to find the eigenvectors.

4. Use the eigenvectors as columns in matrix S.

5. Find the inverse of S (S^-1)

6. Diagonalize the matrix (Λ- capital lambda). Basically, take a matrix of the size that you are using for the problem, fill all spaces with zero's. Then, take your eigenvalues (λ) and insert them in the diagonal going from the first top right space to the last bottom left space. Tadaa! You have Λ !!!

7. Put Λ to whatever power you've been given (in my case, it was 1000 years). Remember that if an eigenvalue is less than 1 but greater than -1, with a high power it goes to zero (-1<λ<1, λ ^1000 = 0)

8. Multiply through the magic formula of S(Λ^n)(S^-1)(initial values vector)