Math 121A: Homework 4-

1. Show that if λ is an eigenvalue of an orthogonal matrix A with eigenvector v, then λ = ±1.

2. Find the eigenvalues and corresponding eigenvectors of the matrix.

3. Consider the system shown below of two masses of mass m, coupled together between two fixed walls via springs with varying spring constants.

Let x(t) and y(t) be the horizontal displacements of the two masses as a function of time.

(a) Write down a system of differential equations for x¨ and y¨.

(b) For the case when m = 1, k₁ = 1, and k₃ = 2, calculate the eigenvalues associated with this system, which are associated with the characteristic frequencies of vibration.

(c) Plot the eigenvalues as a function of k₂ over the range 0 ≤ k2 ≤ 3. Discuss the physical interpretation of the changes to the eigenvalues as k₂ is increased.

4. (a) Let S be the set of solutions y(t) to the differential equation dy/dt = -y fort ≥ 0. With addition and scalar multiplication of elements defined in the usual way, is S a vector space?

(b) Let T be the set of solutions y(t) to the differential equation dy/dt = 1 - y for t ≥ 0. With addition and scalar multiplication of elements defined in the usual way, is T a vector space?