Aufgabe 8.3
a)

A =

2	 1
2	-4

			  		  m
||A||1 = 	max  	  Σ  |a(ij)| = max {2 + 2, 1 + |-4|} = max {2, 5} = 5
		 j=1,..,n  	 i=1

-----------------------------------------------------------------------------

||A||2 = root(λ(max)(A^T * A))

A^T * A =

2	 2	 *	2	 1  =  	2*2+2*2			2*1+2*(-4)		=	 8	 -6
1	-4		2	-4		1*2+(-4)*2		1*1+(-4)*(-4)		-6	  17


det(A^T * A) = (λ - 8)*(λ - 17) - (-6)² = λ² - 25λ + 100


λ1/2 = (25 +- root(625 - 400)) / 2

= (25 +- 15) / 2 = {20, 5}

||A||2 = root(20)

-----------------------------------------------------------------------------

			  		  n
||A||∞ = 	max  	  Σ  |a(ij)| = max {2 + 1, 2 + |-4|} = max {3, 6} = 6
		 i=1,..,m  	 j=1


b)

cond(A)p = ||A||p * ||A^(-1)||p

A^(-1) =

2	 1
2	-4

cond(A)∞ = max {3, 6} * max {4, 5} = 6*5 = 30


Aufgabe 8.4

A =

 2	 1	 1
 4	 3	 4
-6	-5	-6


b =

-1
 2
-2

-------------------------------
L1 =
 1	 0	 0
-2	 1	 0
 3	 0	 1

-------------------------------
L1 * A =
 1	 0	 0	*	 2	 1	 1
-2	 1	 0		 4	 3	 4
 3	 0	 1		-6	-5	-6

=
 2	 1	 1
 0	 1	 2
 0	-2	-3

-------------------------------
L2 =
 1	 0	 0
 0	 1	 0
 0	 2	 1

-------------------------------
L2 * L1 * A =
 1	 0	 0	 * 	 2	 1	 1
 0	 1	 0		 0	 1	 2
 0	 2	 1		 0	-2	-3

=
 2	 1	 1
 0	 1	 2
 0	 0	 1

L2*L1*A = R
<=> L1*A = L2^(-1)*A
<=> A = L1^(-1)*L2^(-1)*R

L1^(-1)*L2^(-1) = L

-------------------------------
L1^(-1) =
 1	 0	 0
 2	 1	 0
-3	 0	 1

-------------------------------
 L2^(-1) =
 1	 0	 0
 0	 1	 0
 0	-2	 1

-------------------------------
L1^(-1)*L2^(-1) =
 1	 0	 0
 2	 1	 0
-3	-2	 1

= L

-------------------------------
L*R =
 1	 0	 0
 2	 1	 0
-3	-2	 1

 *

 2	 1	 1
 0	 1	 2
 0	 0	 1

 =
 2	 1	 1
 4	 3	 4
-6	-5	-6

= A


Ax = b <=> LRx = b

sei Rx = y

1) Ly = b

  y1 + 	  0	  +  0	 =  -1
 2y1 +    y2  +	 0	 =   2
-3y1 +	-2y2  +	 y3  =  -2

y1 = -1
y2 =  4
y3 =  3

2) Rx = y

 2x1 + x2 +	 x3 = -1
  0	 + x2 +	2x3 =  4
  0	 + 0  +	 x3 =  3

x1 = -1
x2 = -2
x3 =  3

	(-1)
x = (-2)
	( 3)


einsetzen in A

 2x1 + 	 x2 +   x3 = (-2) +   (-2) +    3  =  -1
 4x1 +  3x2 +  4x3 = (-4) + 3*(-2) +  4*3  =   2
-6x1 + -5x2	+ -6x3 =   6  +     10 + (-18) =  -2