BP203: Foundations of Mathematical Biology, Fall 2002

Homework

· Problem Set 1. Due Friday 9/27. SOLUTIONS.

o Miller Notes, Problems 1-12.

o Work through the Matlab Getting Started tutorial (don’t worry about turning in anything).

· Problem Set 2. Due Friday 10/4. SOLUTIONS.

o Miller Notes, Problems 13,14,15,17,18,19,20,21,22,25,26

o Matlab Assignment. These problems will use Matlab to solve concrete examples, based on ideas in Ken’s notes. For an answer just write down what the problem asks for, not Matlab output. Also note, I will use Matlab declarations and notation. If you don’t understand what I’m saying, please do the tutorial.

19*. Suppose we wish to diagonalize the matrix M = [0 1 0; 1 0 0; 0 0 1]. Following Ken’s exercise 18, we need to find some orthogonal transformation matrix O with rows that are the orthonormal eigenvectors. First type help eig and read how the function eig works. Notice that [O,D] = eig(M) will return the transformation matrix O and the diagonalized matrix D. What is O for our particular M? Now type O(:,1)’*O(:,1) then O(:,1)’*O(:,2), and finally O(:,1)’*O(:,3). What property of the eigenvectors does this demonstrate? Now type O*M*O’. What matrix is this? Now type O’*D*O. What matrix is this?

21*. Try checking your answer for Problem 21 with the functions eig and eigs. What are the differences between the functions?

22*. To build some intuition for the problem, let’s follow Ken’s hint. Type help poly and see how poly works. What is the characteristic polynomial for M and for M + 6I (Use the command eye(3) to make the 3by3 identity matrix)? Now type help roots. Use roots to find the zeroes of the previous characteristic polynomials. What are the eigenvalues and eigenvectors for both matrices?