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Rechnen mit Matrizen

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by

Martin Weckerle

on 8 November 2018

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Transcript of Rechnen mit Matrizen

(
)
+
(
)
=
)
+
)
(
(
3 1 2
5 3 1
)
)
(
(
1 2 3
4 5 6
+
=
1
6
1
4
5
3
2
4

3

5
9

8

7
)
(
=
Entsprechende Stellen
in den Matrizen werden addiert!
12
2 x 3
2 x 2
3 x 2
3 x 2
Matrizen multiplizieren
Matrizenpotenz
2 4
7 8
3 8
2 4
6 9
9 3
1 4
2 5
7 3 2
1 4 8
3+2
=
2
1
6+1
4+5
2
7
(
)
9
7
3 8
2 4
6 9
9 3
1 4
2 5
2 3
4 1
5 1
2 6
(
)
(
)
.
_______________
_______________
=
2 3
4 1
5 1
2 6
5 1
2 6
2 3
4 1
Rechne:
Zeile mal Spalte
!
Übung: Zeile mal Spalte!
2 3 2
1 2 1
3 1 3
.
1 2 3
2 3 1
1 2 3
(
)
(
)
=
______________________
_______________________
1 4 2
3 5 4
3 1 4
5 6 2
.
(
)
)
(
=
_____________
_____________
1 4 2
3 5 4
3 1 4
5 6 2
1 3
.
1
3
4
5
+
4 5
.
2
+
2
.
?
Spaltenanzahl
1. Matrix
= Zeilenanzahl
2. Matrix
3 4 2
1 2 5
1 2 3
5 6 4
3 7 8
(
)
(
)
.
A B =

.
B A =
.
1 2 3
5 6 4
3 7 8
(
)
3 4 2
1 2 5
(
)
.
3 x 3
2 x 3
5
3
5+3
2 x 3
3 x 3
3
3
3
(
)
1+3
=
2+1
4
3
2
?
Nur Matrizen
gleicher
Größe!
2 4
7 8
7 3 2
1 4 8
2 3
4 1
5 1
2 6
5 1
2 6
2 3
4 1
2
5
2

5
3

2
.
.
3
2
+
=
16
16
5
2
2 3
2

3

1
6
2

1
+
3

6
.
.
=
20
20
1
6
2 3
4 5
+
1 2

=
22
22
4

1
5
2
.
.
5
2
4 1
10
1
6
4

1
16 20
22 10
(
)
1
6
4 1
3
2
2 3 2
1 2 1
3 1 3
2 3 2
1 2 1
3 1 3
1 2 3
2 3 1
1 2 3
1 2 3
2 3 1
1 2 3
1 2 3
2 3 1
1 2 3
1 2 3
2 3 1
1 2 3
2 3 2
1 2 1
3 1 3
2 3 2
1 2 1
3 1 3
3 4 2
1 2 5
1 2 3
5 6 4
3 7 8
A
B
.
.
(
)
(
)
B
A
.
(
)
(
)
1 2 3
5 6 4
3 7 8
3 4 2
1 2 5
.
A B = B A
kommutativ
.
.
2
3
= 2 2 2
= 8
.
.
A
3
= A A A
.
.
A =
2 0
0 1
(
)
2 0
0 1
(
)
2 0
0 1
(
)
2 0
0 1
(
)
=
=
.
.
)
(
2 0
0 1
(
)
.
{
=
8 0
0 1
(
)
A =
3
nur

quadratische

Matrizen!
2 0
0 1
2 0
0 1
4 0
0 1
2 0
0 1
A A = A
(A ) = A
n
n
n+m
n m
m
m
.
.
Rechnen
mit Matrizen
Matrizen addieren
s - Multiplikation
Jeder Eintrag
wird mit der Variablen multipliziert
1 2
3 4
r
(
)
.
=
.
.
.
.
(
)
r
r
r
r
r
1
2
3
4
Matrix - Vektor -
Multiplikation
1
2
3
1 2 3
3 1 2
(
)
(
)
.
Rechne wie bei

zwei

Matrizen!
1
2
3
1 2 3
=
(
1 + 4 + 9
Vektor
15
1 2 3
3 1 2
1
2
3
15

3 + 2 + 6
11
)
3 1 2
11
1
2
3
2

3

2
1
2
1
NR:
2
+
6
+
2
=
10
10
2 3 2
1
2
1
2

3

2
2
3
2
4
+
9
+
4
=
17
17
2 3 2
2
3
2
6
+
3
+
6
=
2

3

2
3
1
3
15
15
2 3 2
3
1
3
6
10
8
8
15
19
10 17 15
(
)
6
10
15
19
8
8
?
?
?
?
?
(
)
1 2 3
3 1 2
)
(
1
2
3
(
)
15
11
Zeile mal Spalte!
.
=
1 2 1
2 1 2
2 1
1 2
2 1
6 6
9 6
(
)
1 2 1
2 1 2
.
)
(
2 1
1 2
2 1
Lineares Gleichungssystem
1 2 1
2 1 2
(
)
.
x
x
x
(
)
1
2
3
=
(
)
x + 2x + x
2x + x + 2x
1
2
3
1
2
3
Vektor
A
.
x
=
b
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