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Unformatted text preview: Department of Mathematical Sciences University of Delaware Prof. T. Angell November 25, 2009 Mathematics 351 Exercise Sheet 8 Exercise 36: Show that if the matrices P and M commute, i.e. if P M = M P then exp (M + P ) = exp (M ) exp (P ). Find an example for which this latter identity is not true. HINT: You will need to look in some calculus text (almost any will do) to ﬁnd out how one computes the product of two series. It is called the “Cauchy product”. Exercise 37: Suppose that there is a nonsingular matrix S so that S −1 AS = D where D is a diagonal matrix. (a) Show that the matrix A has n linearly independent eigenvectors. (b) Find the relationship between the matrices exp A and exp D. Exercise 38: Consider the matrix ￿ 1 9 −1 −5 ￿ (a) Find a fundamental matrix for the system x = Ax. ˙ (b) Find the fundamental matrix exp (A t). Exercise 39: For the matrix ￿ ￿ ￿￿ ￿ ￿ 21 20 01 = + −1 2 02 −1 0 show that the two matrices on the right hand side commute and compute exp (A t). ￿ ￿ −2 1 Exercise 40: Let A = . 1 −2 (a) Show that a fundamental matrix for the homogeneous system x = A x is ˙ ￿ ￿ e−3t e−t X (t) = . −e−3t e−t (b) Find the general solution of the initial value problem for the non-homogeneous system x = Ax + f (t) where f (t) = (2e−t , 3t)￿ with initial condition x(0) = xo . ˙ Due Date: Friday, December 4, in class ...
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