Mathematics 118A, Winter 1998

Midterm Exam I

1. The norm of a linear transformation A: ℝ^{n} → ℝ^{n} is defined

∥A∥ = sup_{\begin{matrix}x∈ℝ^{n} \\ x≠0 \end{matrix}} \frac{∥Ax∥}{∥x∥}.

(a) Prove that ∥A + B∥ ≤ ∥A∥ + ∥B∥.

(b) Show that one can also write ∥A∥ = sup_{∥x∥=1} ∥Ax∥.

(c) Compute ∥A∥ for A = (\begin{matrix}1 & 0 \\ 0 & 2 \end{matrix}) .

Hint: Use part (b). How can you write any unit vector in ℝ^{2}? (4+3+3)

2. Compute the matrix exponential exp(At) for the matrix

A = (\begin{matrix}−1 & −1 \\ 1 & −3 \end{matrix}) .

Hint: The Jordan normal form of A is A = T JT^{−1} with

T = (\begin{matrix}1 & 0 \\ 1 & −1 \end{matrix}) and J = (\begin{matrix}−2 & 1 \\ 0 & −2 \end{matrix}) .

What is the relationship between exp(At) and exp(Jt)? (10)

3. Find all equilibrium points of the system

ẋ = y − x^{2}
ẏ = 1 − y

and determine their stability. Sketch the phase portrait. (You may wish to compute some of the eigenvectors of the linearization to determine the directions of stable and unstable manifolds.) (10)

4. Consider the second order equation

ẍ = −x^{3}.

(a) Find the equilibrium point of the flow. (There is only one.)

(b) Show that the linear analysis of the stability of the critical point is inconclusive.

(c) Find a first integral corresponding to the energy of the system.

(d) At the equilibrium point, does the energy have a local minimum, a local maximum, or a saddle point? What does this tell you about the (nonlinear) stability of the system?

(e) Is the flow volume preserving? (2+2+2+2+2)