25B MATH PROBLEM SET #6 DUE TUESDAY MARCH 22ND Half of this assignment will be graded by Yan and the other half will be graded by Toly. Please turn in the problems from section 1 (which will be graded by Yan) separately from the problems from section 2 (which will be graded by Toly). Standing assumptions For this problem set, unless otherwise stated, you should work over the field C and assume that all vector spaces are finite-dimensional. 1. Yan's problems (1) Computing things (a) Compute the Jordan canonical form of -3 2 0 -6 0 3 0 0 2 1 3 2 6 -3 0 9 (b) Let 2 1 1 A = 0 2 1 0 0 -1 Find a matrix B in Jordan canonical form and an invertible matrix Q such that B = QAQ-1 . (c) Are the matrices 0 2 0 2 1 0 2 0 0 0 2 0 and 0 0 2 0 0 2 similar? (2) Let T : V V be a linear transformation. Suppose that is an eigenvalue of T and that p is a polynomial. (a) Is p() an eigenvalue of p(T )? (b) Are all eigenvalues of p(T ) of this form? (3) Let A and B be n n matrices. Show that every non-zero eigenvalue of AB is an eigenvalue of BA, and conversely.
(4) The set of invertible matrices is open (twice) (a) Show that the determinant of an n n matrix gives a continuous function from 2 2 Rn to R. Deduce that the set of invertible matrices is open in Rn . (b) State and prove a formula for the inverse of the matrix I -A which is valid when every entry of the matrix A is sufficiently small. Use this to 2 give another proof that the set of invertible matrices is open in Rn . Do the 1 1 matrix case first. In fact the set of invertible matrices is also dense in the set of all matrices. 2. Toly's problems (1) Matrix differential equations (a) Show that if A is an n n matrix, x1 , . . . xn are constant scalars, and y denotes the derivative of y with respect to t then x1 y1 . = exp(At) . . . . . yn xn is a solution to the matrix differential equation y1 y1 . = A . . . . . . yn yn (b) Suppose that y1 . . .
()
yn is any solution to (). Compute the derivative of y1 . exp(-At) . . yn with respect to t. Deduce that there exist constants x1 , . . . , xn such that x1 y1 . = exp(At) . . . . . . xn yn So the general solution of the system of linear differential equations () is as in (a). We can compute this efficiently by diagonalising A.
(2) Higher-order linear differential equations and systems of linear equations (a) Suppose that the complex-valued function f (t) satisfies the differential equation f (n) + an-1 f (n-1) + . . . + a1 f (1) + a0 f = 0 where f (i) denotes the ith derivative of f . Write down a matrix-valued differential satisfied equation by the vector f f (1) . . . . f (n-1) (b) Use this to find all solutions to the differential equation d2 f = -f. dt2 (c) Show that the general real-valued solution to this differential equation is f (t) = A cos(t) + B sin(t) for some constants A, B R. This gives an algorithmic way to solve any linear n-th order differential equation with constant coefficients. (3) Jordan blocks and resonance Use the method above to find all real-valued solutions to the differential equations y - 2y - 3y = 0 and y - 2y + y = 0. (4) Autonomous dynamical systems In this problem we work in R2 , and so work over the field R. (a) Suppose that () x y =A x y
where A is a 2 2 matrix with real eigenvalues, one strictly positive and one strictly negative. Show that solution trajectories t (x(t), y(t)) near the origin look like
What are the two straight lines on this picture? This picture is called the "phase portrait" of the autonomous dynamical system (). It is called "autonomous" because the coefficients of () -- the entries of A -- do not vary with time. (b) Suppose that we change A. What are the other possible phase portraits? There are lots of possibilities. A could have both eigenvalues positive, or two complex eigenvalues, or . . . (5) A simple model of an epidemic A simple model of an epidemic in a city is as follows. Susceptible people enter the city at a constant rate of per day. Infected people recover or die after a certain number of days. If they recover, they are immune. The num...