Math410 (Bayly) Homework 6 (due Monday 5 March)

Section 5.3: All except econ applications; turn in 5, 7, 8, 10, 16.

Section 5.4: All except a few at the end which I forget; turn in 4, 8, 13, 15, 20.

EXTRA CREDIT

Alternative method for calculating matrix exponentials that works whether or not the matrix is diagonalizable.

(1) Recall the Cayley-Hamilton theorem says that if A is an n x n matrix, with , then , the matrix of all zero entries. Argue that this implies you can express as a scalar linear combination of .

(2) Argue also that you can express any higher power of A as a scalar linear combination of . Therefore (continue to give your reasoning) the matrix exponential , where are scalar functions of t.

(3) Now we just have to find out what the functions are. To do this, use the fact that the matrix exponential satisfies , and so on. Therefore if each satisfies the equation below, we’ll be all set: .

(4) Last but not least, we need initial conditions for the functions. These are (can you say why?): , for each p from 0 to n-1.

(5) Use this method to find for the nondiagonalizable matrix , and also the diagonalizable matrices and . Compare with the “old” method for diagonalizable matrices.