Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Engineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 14, February 4/5, 2008 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Find the eigenvalues and the eigenvectors of the following matrices

A₁ =

5 −3

3 5

, A₂ =

5 −3

−3 5

, A₃ =





5 2 0 2 5 0 0 0 3



.

When possible, find an orthonormal system of eigenvectors, e. g.C_i = (~c⁽ⁱ⁾₁ , ~c⁽ⁱ⁾₂ , ~c⁽ⁱ⁾₃ ), ofA_i (i= 1,2,3). Find also C_i^TA_iC_i.

2. Let the following matrix be given:

A=



 1/√

2 −1/√

2 0

1/√

2 1/√

2 0

0 0 1/√

2



.

a) Find the eigenvalues of A and the eigenvectors corresponding to the real eigenvalues.

b) Describe the mapping corresponding to A geometrically.

3. Find the matrix representations to the following mappingsf :R³ →R³ and check whether they are orthogonal.

a) Scaling in x-,y- andz-direction by a factor of 2.

b) Addition of the vector~a = (3,0,0)^T.

c) Counterclockwise rotation about the z-axis by π/2.

d) Reflection with respect to the plane which contains the origin and is perpen- dicular to the vector (1,1,1).

Homework

1. Find the eigenvalues and the eigenvectors of the following matrices

A1 =

0 1 6 −5

, A2 =

2 1 3 0

, A3 =





0 1 0

0 0 1

1 −3 3



.

When possible, find an orthonormal system of eigenvectors, e. g.C_i = (~c⁽ⁱ⁾₁ , ~c⁽ⁱ⁾₂ , ~c⁽ⁱ⁾₃ ), ofA_i (i= 1,2,3). Find also C_i^TA_iC_i.

2. Let the following matrices be given:

A=





0 −1 0

1 0 0

0 0 2



, A =





0 1 0 1 0 0 0 0 2



.

a) Find the eigenvectors and eigenvalues of A and B.

b) Describe the mappings corresponding to A and B geometrically.

3. Find the matrix representations to the following mappingsf :R³ →R³ and check whether they are orthogonal.

a) Scaling in y-direction by a factor of ¹₂ and inz-direction by a factor of 2.

b) Addition of the vector~a = (1,1,1)^T.

c) Counterclockwise rotation about they-axis by π/3.

d) Reflection with respect to the x-y-plane.

4. Compute the LU-factorization of the following matrix:

A=





5 6 7

10 11 12 15 14 14



.

Solve the SLE A~x = ~b for the right-hand sides ~b₁ = (18,33,43)^T and ~b₂ = (−1,−1,1)^T.

5. Compute QR-factorization of the following matrix:

A=

1 −1

1 1

.

Solve the SLEA~x=~bfor the right-hand sides~b1 = (−1,3)^T and~b2 = (−1,7)^T. 6. a) Find all complex solutions of the equation

z⁶−7z³−8 = 0. b) For the complex number

z₀ = 3 +i i−1 find |z₀|, z₀ and the Cartesian form of z₀+ 3.

c) Find the Euler form ofz1 =√ 3 + 3i.