Exercise 1&2 - Consider the square matrix

30449 Mathematics II (Applied) —BESS Assignment 3

Exercise 1 Consider the square matrix

A =

? ? 1 2 10 1 0 1 0 1

? ?

(a) Calculate the eigenvalues of A and decide whether the matrix is diagonalizable. Justify your answer. (b) Find the associated eigenspaces of A. (c) Write the eigendecomposition of the power matrix A3 and of the exponential matrix eA. (d) Consider the matrix

B = A ·eA

Prove B is diagonalizable and find the diagonal matrix D similar to B.

Exercise 2 Let b : R → R. Consider the dynamical system

x′ = x2b(t)

(a) Provide a suffi cient condition for the function b : R → R such that for every initial point (x0, t0) ∈ R2 there exists a unique solution of the IVP. Justify your answer. (b) Putting

b(t) = et

compute the general solution ?(t;c) of the ODE. (c) Find the particular solution ?? of the IVP with starting point (x0, t0) = (0,1) and specify its domain I ⊆ R.