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.
![638_Figure.png](https://secure.tutorsglobe.com/CMSImages/638_Figure.png)
3. Consider the system shown below of two masses of mass m, coupled together between two fixed walls via springs with varying spring constants.
![1024_Figure1.png](https://secure.tutorsglobe.com/CMSImages/1024_Figure1.png)
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?