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Brandenburg Technical University Cottbus
Department 1, Institute of Mathematics Chair for Numerical Mathematics an Scientific Computing Prof. Dr. G. Bader, Dr. A. Pawell

Problem Session to theCourse: Mathematics I
Environmental and Resource Management WS 2002/03 Solutions to Sheet No. 13 (Deadline: January, 27/28 2002)

Homework
H 13.1: Eigenvalues of A: λ1 = −1, λ2/3 = 1 Eigenvectors: λ1= −1:   2 0 0  0 1 1  x = 0, 0 1 1 λ2/3 = 1: 1 c2 = √ (0, 1, 1)T , 2  C= 0
1 √ 2 1 − √2

x = (0, 1, −1)T ,

1 c1 = √ (0, 1, −1)T 2

c3 = (1, 0, 0)T

1 √ 2 1 √ 2

 0 1 0 . 0

q(x)= xT Ax = x2 + 2x2 x3 . 1 B = C, C T AC = diag(1, −1, −1)

2 2 2 q(Cy) = (Cy)T A(Cy) = y T C T ACy = y T diag(1, −1, −1)y = y1 − y2 − y3 .

H 13.2:

1−λ det(A − λE) = det  0 0



1 1−λ −1 1 = 5 −1 − λ

(1 − λ)[(1 − λ)(−1 − λ) + 5] = (1 − λ)[4 + λ2 ] = 0



λ1 = 1,

λ2/3 = ±2i

Eigenvectors:  1−λ  0 0 λ1 = 1:  1 1−λ −1  1 x = 0 5 −1 − λ

 0 1 1  0 0 5 , 0 −1−2

x = t(1, 0, 0)T

λ1 = 2i: 

 1 − 2i 1 1  x = 0 0 1 − 2i 5 0 −1 −1 − 2i 2i , −1 − 2i, 1 1 − 2i
T



 1 − 2i 1 1  0 −1 −1 − 2i  x = 0, x = 0 0 0 λ1 = −2i:   1 + 2i 1 1  x = 00 1 + 2i 5 0 −1 −1 + 2i   1 + 2i 1 1  0 −1 −1 + 2i  x = 0, x = 0 0 0 H 13.3: det(A − λE) = det 1−λ 1 1 3−λ

−2i , −1 + 2i, 1 1 + 2i

T

= (1 − λ)(3 − λ) − 1 = λ2 − 4λ + 2.

Eigenvalues:λ1/2 = 2 ± √ 2 √ 2 x = 0, x= √ (−1 + 4−2 2 1 √ 2, 1)T

Eigenvectors: λ = 2 + −1 − √ √

2 √1 1 1− 2

λ=2−

2 √ 2 √1 1 1+ 2 x = 0, x= √ (−1 − 4+2 2 1 √ 2, 1)T

−1 +

2

 B

2 √ 4−2 2 √1 √ 4−2 2

−1+ √



−1− 2 √ √



  √ 2+ 2 0



4+2 2 1 √ 4+2 2

B −1 = B T ,

B −1 AB =

0√ 2− 2

Additional Problems
P 13.1: 5−λ 2 det(A − λE) = det  0   2 0 5−λ 0 = 03−λ

(3 − λ) (5 − λ)2 − 4 = λ2 − 10λ + 21 = 0 ⇒  λ1 = 7, λ2,3 = 3.   1 1   1 x1 = √ 2 0  0 x3 =  0  1 

 −2 2 0  2 −2 0  x = 0, 0 0 −4   2 2 0  2 2 0  x = 0, 0 0 0

  1 1 ...
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