Exercises 36-44
Exercise 45.

Solve the pair of differential equations

 x'1 (t) = x1 (t) + x2 (t) x'2 (t) = - x1 (t) + x2 (t)

with initial conditions x1(0) = x2(0) = 1000.

Exercise 46.

Solve the initial value problem

x''(t) -3x'(t) - 10x(t) = 0;   x(0) = 3,   x'(0) = 2,

by converting it into a system of differential equations of order 1 and then solving the system.

Exercise 47.

Use Gershgorin discs to study the location of the eigenvalues of the matrix

 2-j 2 1 1 2j 0 j 1 2
Draw a picture.
Exercise 48.

Show that if is an eigenvalue of the matrix

 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2
then 0 < < 4.
Exercise 49.

Consider the matrix

 A = 2 -2 1 -1 3 -1 -2 -4 3
Compute an approximation of the largest eigenvalue of A. (Compute 1(4)). What are the corrresponding eigenvectors?
Exercise 50.

Compute the norms ||A||1 ,  ||A|| and ||A||Fr of the matrix

 A = -1 1 0 -1 0 1 -2 1 1 0 1 -1 -2 0 -1 1

Exercise 51.

Consider the matrix A of the previous exercise. Compute ||A||2 iteratively. Start from the vector 0 = (0, 1, - 1, 1).

Exercise 52..

Compute the condition number of A with respect to 1-norm and -norm. Here

 A = 0.501 -0.343 . -0.872 0.597

Exercise 53.

Compute eA, where
 a) A = 3 -2 , -1 4
 b) A = 0 1 0 0 . 0 0 1 0 0 0 0 1 0 0 0 0
An example

Exercise 54.

Compute the spectral radius of the matrix

 A = 1 0 0 0 1 -1 -1 -1 2
Exercises 55-60
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All exercises