Download Chapter 3

Chapter 8
Linear Algebraic Equations
and Matrices
Three individuals connected by bungee cords
Free-body diagrams
Newton’s
second law
 m1 g  k 2 ( x 2  x 1 )  k 1 x 1  0

 m2 g  k3 ( x 3  x 2 )  k 2 ( x 2  x 1 )  0
m g  k ( x  x )  0
3
3
2
 3
Rearrange the equations
[K] {x} = {b}
( k1  k2 ) x1

 k2 x1



 k2 x 2
( k2  k3 ) x 2
 k3 x 2
 k3 x 3
 k3 x 3
 m1 g
 m2 g
 m3 g
Newton’s second law – equation of motion
Mass-spring
system
(similar to
bungee jumpers)
Kirchhoff’s current and voltage rules
Resistor
circuits
Linear Algebraic Equations
Solved single equations previously
f x  0
Now consider more than one variable and more
than one equation
f 1  x1 , x 2 ,..., xn   0
f 2  x1 , x 2 ,..., xn   0

f n  x1 , x 2 ,..., xn   0
Linear Systems
Linear equations and constant coefficients
a11 x1  a12 x 2  ...  a1n xn  b1
a 21 x1  a 22 x 2  ...  a 2 n xn  b2

an1 x1  an 2 x 2  ...  ann xn  bn
aij and bi are constants
Forces on a Truss
Most obvious example in Civil Engineering
trusses: force balance at joints
a11 F1  a12 F2  ...  a1 n Fn  R1

a 21 F1  a 22 F2  ...  a 2 n Fn  R2


an 1 F1  an 2 F2  ...  ann Fn  Rn
R
F1
F2
F3
Mathematical background
It is convenient to write system of equations
in matrix-vector form
[ A][ x ]  [b]
 a11
a
 21
 

 
an 1
or
[ A]{ x }  { b}
a12  a1 n   x1   b1 





a 22  a 2 n   x 2  b2 

       
   

    
an 2  ann   xn  bn 
Matrix Notations
Column 4 (second index)
 a11
a
 21
 a31
A  
a
 41
 

 am 1
a12
a22
a32
a42

am 2
a13
a23
a33
a43

am 3
a14
a24
a34
a44

am 4






a1 n 
a2 n 

a3 n 

a4 n 
 

amn 
Row 3
(first index)
Scalars, Vectors, Matrices
MATLAB treat variables as “matrices”
Matrix (m  n) - a set of numbers arranged in
rows (m) and columns (n)
Scalar : 1  1 matrix
Row Vector : 1  n matrix ( [b] or b )
Column Vector : m  1 matrix ( [c] or {c} )
[a]  5.27 
[b]  b  5.02  2.3 7.21
 5.02 
[c ]  {b}  [b]T    2.3


 7.21
1
3 2
5

[ D]    2
4 3.2 9.5 


  0.5  1 7.2  2 
Square Matrix
Square matrix, m = n
Particularly important when solving simultaneous
equations in engineering applications

 a11
a
21

[ A] 
a31

 a41
a12
a22
a32
a13
a23
a33
a42
a43
aii – principle or main diagonal
a14 
a24 

a34 

a44 
Matrix Operations

Transpose
 a11
a
[ A]   21
a31

 a41
a12
a22
a32
a13
a23
a33
a42
a43
a14 
a24 

a34 

a44 
 a11
a
[ A]T   12
 a13

a14
a21
a22
a23
a31
a32
a33
a24
a34
In MATLAB, transpose is A
Trace is sum of diagonal elements
In MATLAB, trace(A)
a41 
a42 

a43 

a44 
Matrix Transpose
x  4
 2 3 ;
y  3 1  2 
 4
x   2 ;
 
 3 
 4
x  * y    2  3 1  2  
 
 3
x * y  4
 3
y'   1
 
  2 
4  8
 12
 6  2
4


 9
3  6 
 3
 2 3  1  ( 4 )( 3 )  ( 2 )( 1)  3( 2 )  4
 
  2 
Special Matrices
• symmetric matrices
a
b
[ A]  
c

d
b
e
f
g
c
f
h
i
2 1
 5
 2 2
8

[ B]    1
8
3

2
 4 5
 6
1
0
d
g

i

j
4
6
5
1

2
0

1
3
3  4 
aij = aji
T
[A] = [A]
Special Matrices
• Diagonal matrix
a11

[ A]  



a22
a33





a44 
• Identity matrix
1

 1


[I ]  

1 


1

[A][I] = [I][A] = [A]
Special Matrices
 Banded matrix – all elements are
zero, with the exception of a band
centered on the main diagonal
 a11
a
21

[ A] 



a12
a22
a32
a23
a33
a43


Tridiagonal – three
 non-zero bands
a34 

a44 
Special Matrices
 a11
a
a22
21
[ A]  
a31 a32

 a41 a42
• upper triangular
a11 a12

a22
[ A]  



• lower triangular
a33
a43
a13
a23
a33





a44 
a14 

a24

a34 

a44 
Matrix Operation Rules
Matrix identity
[A] = [B] if and only if aij = bij for all i and j
Matrix Addition and Subtraction
[C] = [A] + [B]  Cij = Aij + Bij
[C] = [A]  [B]  Cij = Aij  Bij
Addition and Subtraction
Commutative
[A] + [B] = [B] + [A]
[A]  [B] = [B] + [A]
Associative
( [A] + [B] ) + [C] = [A] + ( [B] + [C] )
( [A] + [B] )  [C] = [A] + ( [B]  [C] )
( [A]  [B] ) + [C] = [A] + ([B] + [C] )
Multiplication of Matrix by a Scalar
 a11
a
[ A]   21
a31

a41
a12
a22
a32
a13
a23
a33
a42
a43
a14   1
a24   4

a34   5
 
a44    2
 ga11
 ga
[ B]  g[ A]  [ A]g   21
 ga31

g=5
 ga41
ga12
ga22
ga32
ga13
ga23
ga33
ga42
ga43
3 2
2
2
3
4
1  3

7
0

1
6
ga14   5
ga24   20

ga34   25
 
ga44    10
15  10
10
10
15
20 
5  15 

35
0

5
30 
Matrix Multiplication
 Visual depiction of how the rows and columns line up
in matrix multiplication
Matrix Multiplications
Recall how matrix multiplication works
a
d

b
e
g j
c 
 ag  bh  ci

h
k





f
dg  eh  fi

 i l 
g j
 h k  a

 d
 i l  
b
e
 ga  jd
c 
  ha  kd

f
 ia  ld
aj  bk  cl 

dj  ek  fl 
gb  je
gc  jf 
hb  ke hc  kf 
ib  le ic  lf 
[A]*[B]  [B]*[A]
Matrix Multiplication
 Matrix multiplication can be performed only if the
inner dimensions are equal
Matrix Multiplication
Interior dimensions have to be equal
Ai j  B jk  C ik
For a vector
Ai j  x j  bi
We will be using square matrices
Ann  xn  bn
MATLAB
 In Fortran, the matrix multiplication have to be done by Do
Loops
 In MATLAB, it is automatic
A*B = C
 Note no period ‘.’ (not element-by-element operation)
 For vectors
A*x = b
Matrix Multiplication
Associative
( [A] [B] ) [C] = [A] ( [B] [C] )
Distributive
[A] ( [B] + [C] ) = [A] [B] + [A] [C]
([A] + [B] ) [C] = [A] [C] + [B] [C]
Not generally commutative
[A] [B]  [B] [A]
Matrix Inverse
 Matrix division is undefined
 However, there is a matrix inverse for
non-singular square matrices
[A]1 [A] = [A] [A]1 = [I]
 Multiplication of a matrix by the inverse
is analogous to division
Augmentation
a11 a12
a
a 22
21

A
a 31 a 32

a41 a42
a11

a 21

A
a 31

a 41
a12
a 22
a 32
a 42
a13
a 23
a 33
a43
a13
a 23
a 33
a 43
a14 
a 24 
a 34 

a44 
a14
a 24
a 34
a 44
1
0
0
0
0
1
0
0
0
0
1
0
0

0
0

1
 Whatever you do to left-hand-side, do to the righthand side (useful when solving system of equations)
Augmented Matrix
a 11 x 1
a 21 x 1
a 31 x 1
a 41 x 1
 a 12 x 2
 a 22 x 2
 a 32 x 2
 a 42 x 2
 a 11
a
21

 A b  
a 31

a 41
 a 13 x 3
 a 23 x 3
 a 33 x 3
 a 43 x 3
 a 14 x 4
 a 24 x 4
 a 34 x 4
 a 44 x 4
a 12
a 13
a 14
a 22
a 32
a 42
a 23
a 33
a 43
a 24
a 34
a 44
 b1
 b2
 b3
 b4
b1 

b2 
b3 

b4 
MATLAB Matrix Manipulations
>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]
A =
1
5
3
-4
2
5
6
-1
3
4
-2
5
-1
3
2
6
1
2
3
-1
5
5
4
3
3
6
-2
2
-4
-1
5
6
Create a matrix
>> A'
ans =
Matrix transpose
MATLAB Matrix Manipulations
Matrix Concatenation
>> C = A + B
C =
>> x = [5 1 -2 3];
>> y = [2 -1 4 2];
>> z = [3 -2 1 -5];
>> B = [x; y; x; z]
B =
6
6
1
-1
4
4
10
1
8
5
-4
8
2
1
3
1
( = A)
5
1
-2
3
>> C = C – B
2
-1
4
2
C =
5
1
-2
3
1
5
3
-4
3
-2
1
-5
2
5
6
-1
3
4
-2
5
-1
3
2
6
Addition and Subtraction
MATLAB Matrix Manipulations
 Matrix multiplication and element-by-element operation
>> A*B
A =
ans =
1
5
3
-4
2
5
6
-1
18
7
8
42
3
4
-2
5
47
5
3
39
-1
3
2
6
28
-13
19
-14
29
-14
16
-21
B =
5
1
-2
3
>> A.*B
2
-1
4
2
ans =
5
1
-2
3
5
5
-6
-12
3
-2
1
-5
4
-5
24
-2
15
4
4
15
-3
-6
2
-30
A*B  A.*B
MATLAB Matrix Manipulations
>> D = [2 4 3 1; 3 -5 1 2; 1 -1 3 2]
A =
1
5
3
-4
2
5
6
-1
2
4
3
1
3
4
-2
5
3
-5
1
2
-1
3
2
6
1
-1
3
2
D =
>> A*D
??? Error using ==> *
Inner dimension
must agree
A*D  D*A
Inner matrix dimensions must agree.
>> D*A
ans =
18
45
26
9
-6
0
-19
10
6
18
-5
24
MATLAB Matrix Manipulations
>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]
>> format short; AI = inv(A)
AI =
-0.2324
0.2520
0.1606
-0.2467
0.2428
-0.1397
0.0457
0.1005
-0.1436
0.2063
-0.0862
0.0104
-0.1123
0.0431
0.0326
0.0718
>> A*AI
Matrix Inverse
ans =
1.0000
0
0
-0.0000
-0.0000
1.0000
0
0.0000
0.0000
0
1.0000
-0.0000
0.0000
0
0.0000
1.0000
MATLAB Matrix Manipulations
>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]
>> I = eye(4)
I =
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
>> Aug = [A I]
Aug =
1
5
3
-4
1
0
0
0
2
5
6
-1
0
1
0
0
3
4
-2
5
0
0
1
0
-1
3
2
6
0
0
0
1
>> [n, m] = size(Aug)
n =
4
m =
Matrix Augmentation
8
Bungee Jumpers
Jumper
Mass (kg) Spring constant (N/m) Unstretched cord length (m)
Top (1)
60
50
20
Middle (2)
70
100
20
Bottom (3)
80
50
20
 m1 g  k 2 ( x 2  x 1 )  k1 x 1  0

m2 g  k3 ( x 3  x 2 )  k2 ( x 2  x1 )  0
m g  k ( x  x )  0
3
3
2
 3
[K] {x} = {b}
 k2
0   x 1   m1 g 
( k1  k 2 )

 k2 ( k2  k3 )  k3   x 2    m2 g 

  


0
 k3
k3   x 3   m3 g 
k1 = 50
k2 = 100
stiffer cord
k3 = 50
>> k1 = 50; k2 = 100; k3 = 50;
>> K=[k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3]
K =
150 -100
0
-100
150
-50
0
-50
50
>> format short
>> g = 9.81; mg = [60; 70; 80]*g
mg =
588.6000
686.7000
784.8000
>> x=K\mg
x =
41.2020
55.9170
71.6130
>> x = inv(K)*mg
x =
41.2020
55.9170
71.6130
>> xi = [20; 40; 60];
>> xf = xi + x
xf =
61.2020
95.9170
Final positions of bungee jumpers
131.6130
TRUSS
4
5
F45
F14
H1
F24

1
V1
F35
F25


F12
2

F23
W = 100 kg
3
V3
Statics: Force Balance
Node 1
Node 2
Node 3
Node 4
Node 5
 F y , 1

 F x , 1
 F y , 2

 F x , 2
 F y , 3

 F x , 3
 F y , 4

 F x , 4
 F y , 5

 F x , 5
 V1  F14 sin   0
 H 1  F12  F14 sin   0
 F24 sin   F25 sin   100
  F12  F23  F24 cos   F25 cos   0
 V 3  F35 sin   0
  F23  F35 cos   0
  F14 sin   F24 sin   0
  F14 cos   F24 cos   F45  0
  F25 sin   F35 sin   0
  F25 cos   F35 cos   F45  0
Exampe: Forces in a Simple Truss
1
0

0

0
0

0
0

0
0

0
0 0
1 0
0
0
0
0
0
0
0
0
0
1
sin
cos 
0
0
0
0
0
0
0
0
0
0
0
0
sin 
sin 
0
0 1
0
1
cos 
cos 
0
1
0
0
0
0
0
sin 
0
0
0 1
0
0  cos 
0
0  sin
0  sin 
0
0
0
0  cos 
0
cos 
0
0
0
0
0
0
0  sin 
sin 
0
0
0
0
0
cos 
cos 
0   V1   0 
 
0   H 1   0 
0   V 3  100 
  

0   F12   0 
0   F14   0 
   

0   F23   0 
0   F24   0 
  

1  F25   0 
0   F35   0 
  

 1  F45   0 
Define Matrices A and b in script file
function [A, b]=Truss(alpha, beta, gamma, delta)
A = zeros(10,10);
A(1,1) = 1; A(1,5) = sin(alpha);
A(2,2) = 1; A(2,4) = 1; A(2,5) = cos(alpha);
A(3,7) = sin(beta); A(3,8) = sin(gamma);
A(4,4) = -1; A(4,6) = 1; A(4,7) = -cos(beta);
A(4,8) = cos(gamma);
A(5,3)= 1; A(5,9) = sin(gamma);
A(6,6) = -1; A(6,9) = -cos(delta);
A(7,5) = -sin(alpha); A(7,7)=-sin(beta);
A(8,5) = -cos(alpha); A(8,7) = cos(beta); A(8,10)=1;
A(9,8) = -sin(gamma); A(9,9) = -sin(delta);
A(10,8) = -cos(gamma); A(10,9) = cos(delta); A(10,10) = -1;
b = zeros(10,1); b(3,1)=100;
f = A\b
[A]{ f } = {b}  { f } = [A]1 {b}