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.

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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.

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Exercise 47.

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

2-j 2 1
12j 0
j 1 2
Draw a picture.

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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.

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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?

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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

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Exercise 51.

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

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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

Answer


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
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Exercise 54.

Compute the spectral radius of the matrix

A = 1 0 0
0 1-1
-1-1 2
Answer
Exercises 55-60
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