MATH 395 PROBLEMS 4

IGOR KRIZ

Regular problems:

1. Rewrite the following system of differential equations in the form y′_{i} = f_{i}(x, y_{1}, …, y_{n}), i = 1, …, n:

z′′′′_{1} = g_{1}(x, z_{1}, z_{2}, z′_{1}, z′_{2}, z′′_{1}, z′′_{2}, z′′′_{1}, z′′′_{2}),
z′′′_{2} = g_{2}(x, z_{1}, z_{2}, z′_{1}, z′_{2}, z′′_{1}, z′′_{2}, z′′′_{1}).

2.

(a) Two n × n matrices A, B are called similar if there exists an invertible n × n matrix P such that A = P^{−1}BP. Prove that similar matrices have the same eigenvalues.

(b) Find an explicit formula for the eigenvalues of a 2 × 2 matrix.

3. A normed space is a vector space V over ℝ together with a function (called norm) assigning to every x ∈ V a number ||x|| ≥ 0 such that (1) ||x|| = 0 iff x = 0,

(2) ||x + y|| ≤ ||x|| + ||y||. Prove that if V is a normed space, then V is a metric space with metric

ρ(x, y) = ||x − y||.

Give at least three different examples of norms on ℝ^{n}. (Challenge: can you give infinitely many examples?)

4. An n × n matrix is called a projection matrix if AA = A. Prove an eigenvalue of a projection matrix is either 0 or 1.

Challenge problems:

5. Let ℝ^{n} be a metric space with metric

ρ((x_{1}, ...x_{n}), (y_{1}, ..., y_{n})) = max_{i=1,...,n} |x_{i} − y_{i}|

and let f : ℝ → ℝ be a map such that for some constant K,

ρ(f(x), f(y)) < Kρ(x, y)

for every x, y ∈ ℝ. Prove that there exists a λ ∈ ℝ and an x ∈ ℝ^{n} such that

f(x) = λx.

Is this true if the above metric is replaced by the Euclidean metric?

6. Give an example of a metric d on ℝ^{n} such that (ℝ^{n}, d) is homeomorhpic to (ℝ^{n}, ρ) (ρ is defined in Problem 5), but such that (ℝ^{n}, d) is not complete.