# A, B ```CSE123
Lecture 5
Arrays and Array Operations
Definitions
• Scalars: Variables that represent single numbers. Note that
complex numbers are also scalars, even though they have two
components.
• Arrays: Variables that represent more than one number. Each
number is called an element of the array. Array operations
allow operating on multiple numbers at once.
• Row and Column Arrays (Vector): A row of numbers (called a
row vector) or a column of numbers(called a column vector).
• Two-Dimensional Arrays (Matrix): A two-dimensional table of
numbers, called a matrix.
Vector Creation by Explicit List
• A vector in Matlab can be created by an explicit list, starting with a left
bracket, entering the values separated by spaces (or commas) and
closing the vector with a right bracket.
&gt;&gt;x=[0 .1*pi .2*pi .3*pi .4*pi .5*pi .6*pi .7*pi .8*pi .9*pi pi]
&gt;&gt;y=sin(x)
&gt;&gt;y =
Columns 1 through 7
0 0.3090 0.5878 0.8090 0.9511 1.0000 0.9511
Columns 8 through 11
0.8090 0.5878 0.3090 0.0000
• A vector element is addressed in Matlab with an integer index (also
called a subscript) enclosed in parentheses.
&gt;&gt; x(3)
ans =
0.6283
&gt;&gt; y(5)
ans =
0.9511
Colon notation: Addresses a block of elements. The format is:
(start:increment:end)
Note start, increment and end must be positive integer numbers.
If the increment is to be 1, a shortened form of the notation may be used:
(start:end)
&gt;&gt; x(1:5)
ans =
0 0.3142 0.6283 0.9425 1.2566
&gt;&gt; x(7:end)
ans =
1.8850 2.1991 2.5133 2.8274 3.1416
&gt;&gt; y(3:-1:1)
ans =
0.5878 0.3090
&gt;&gt; y([8 2 9 1])
ans =
0.8090 0.3090 0.5878 0
Vector Creation Alternatives
• Combining: A vector can also be defined using another vector that has
&gt;&gt; B = [1.5, 3.1];
&gt;&gt; S = [3.0 B]
S =
3.0000 1.5000 3.1000
• Changing: Values can be changed by referencing a specific address
&gt;&gt; S(2) = -1.0;
&gt;&gt; S
S =
3.0000 -1.0000 3.1000
• Extending: Additional values can be added using a reference to a specific
&gt;&gt; S(4) = 5.5;
&gt;&gt; S
S =
3.0000 -1.0000 3.1000 5.5000
&gt;&gt; S(7) = 8.5;
&gt;&gt; S
S =
3.0000 -1.0000 3.1000 5.5000 0 0 8.5000
Vector Creation Alternatives
• Colon notation:
(start:increment:end)
where start, increment, and end can now be floating point numbers.
x=(0:0.1:1)*pi
x =
Columns 1 through 7
0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850
Columns 8 through 11
2.1991 2.5133 2.8274 3.1416
• linspace: generates a vector of uniformly incremented values, but instead of
specifying the increment, the number of values desired is specified. The
form:
linspace(start,end,number)
The increment is computed internally, having the value:
end  start
increment 
number  1
Vector Creation Alternatives
&gt;&gt; x=linspace(0,pi,11)
x =
Columns 1 through 7
0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850
Columns 8 through 11
2.1991 2.5133 2.8274 3.1416
logspace(start_exponent,end_exponent,number)
To create a vector starting at 100 = 1,ending at 102 = 100 and having 11 values:
&gt;&gt; logspace(0,2,11)
ans =
Columns 1 through 7
1.0000 1.5849 2.5119 3.9811 6.3096 10.0000 15.8489
Columns 8 through 11
25.1189 39.8107 63.0957 100.0000
Vector Length
length(x): To determine the length of a vector array.
&gt;&gt; x = [0 1 2 3 4 5]
x =
0 1 2 3 4 5
&gt;&gt; length(x)
ans =
6
Vector Orientation
A column vector, having one column and multiple rows, can be created by
specifying it element by element, separating element values with semicolons:
The transpose operator (’) is used to transpose a row vector into a
column vector
&gt;&gt; c = [1;2;3;4;5]
c =
1
2
3
4
5
&gt;&gt; a = 1:5
a =
1 2 3 4 5
&gt;&gt; c = a’
c =
1
2
3
4
5
Matrix arrays in Matlab
Vector
&amp;
Matrix
•Use square brackets.
•Separate elements on the same row with spaces or
commas.
•Use semi-colon to go to the next row.
A  1 2 3
A= [ 1 2 3 ]; A= [ 1, 2, 3 ];
1 2

B  
3 4
B= [ 1 2 ; 3 4 ];
5
 
C  6
7
 
C= [ 5 ; 6 ; 7 ];
Matrix Arrays
A matrix array is 2D, having both multiple rows and multiple columns.
Creation of 2D arrays follows that of row and column vectors:
– Begin with [ end with ]
– Spaces or commas are used to separate elements in a row.
– A semicolon or Enter is used to separate rows.
&gt;&gt;f = [1 2 3; 4 5 6]
f =
1 2 3
4 5 6
&gt;&gt; g = f’
g =
1 4
2 5
3 6
&gt;&gt; h = [1 2 3
4 5 6
7 8 9]
h =
1 2 3
4 5 6
7 8 9
&gt;&gt; k = [1 2;3 4 5]
??? Number of elements in each
row must be the same.
Special matrix creation
Manipulations and Combinations:
A=
Matrix full of 10:
&gt;&gt; A=10*ones(2,2)
Matrix of random
numbers between
0 and 10
&gt;&gt; B=10*rand(2,2)
Matrix of random
numbers between
-1 and 0
&gt;&gt; C= -rand(2,2)
Matrix of random
numbers between
-1 and 1
&gt;&gt; C=2*rand(2,2) –ones(2,2)
B=
10
10
10
10
4.5647
8.2141
0.1850
4.4470
C = -0.4103 -0.0579
-0.8936 -0.3529
&gt;&gt; C=2*rand(2,2) -1
C = -0.6475
0.8709
-0.1886
0.8338
Special matrix creation
Concatenation:
Combine two (or more) matrices into one
Notation:
Square
brackets
C=[ A, B ]
&gt;&gt; A=ones(2,2);
&gt;&gt; B=zeros(2,2);
&gt;&gt;C=[A , B]
&gt;&gt;D=[A ; B]
D=
C=
1
1
1
1
0
0
1
1
1
1
0
0
0
0
0
0
Matrix indexation
Obtain a single value from a matrix:
Notation:
1 2 3


A  3 2 1
1 2 4


Ex:
A(2,1)
Row index
want to know a21
&gt;&gt; A=[1 2 3; 3 2 1; 1 2 4];
&gt;&gt; A(2,1)
&gt;&gt; A(3,2)
ans =
ans =
3
2
Column index
Matrix indexation
Obtain more than one value from a matrix:
Notation:
Colon defines a
“range”: 1 to 10
Ex: X=1:10
Colon
&gt;&gt; A=[1 2 3; 3 2 1; 1 2 4];
Row 1 to 3
A(1:3,2:3)
1 2 3


A  3 2 1
1 2 4


Column 2 to 3
Colon can also be used as a “wildcard”
&gt;&gt; B=A(1:3,2:3)
B=
2
3
2
1
2
4
Row 2, ALL
columns
&gt;&gt; C=A(2,:)
C=
3
2
1
Matrix size
Command
Description
s = size(A)
For an m x n matrix A, returns the two-element row
vector s = [m, n] containing the number of rows and
columns in the matrix.
[r,c] = size(A)
[r,c] = size(A) Returns two scalars r and c containing
the number of rows and columns in A, respectively.
r = size(A,1)
Returns the number of rows in A in the variable r.
c = size(A,2)
Returns the number of columns in A in the variable c.
Matrix size
&gt;&gt; A = [1 2 3; 4 5 6]
A=
123
456
&gt;&gt; s = size(A)
s=
23
&gt;&gt; [r,c] = size(A)
r=
2
c=
3
&gt;&gt; whos
Name Size Bytes Class
A
2x3
48
double
ans
1x1
8
double
c
1x1
8
double
r
1x1
8
double
s
1x2
16
double
array
array
array
array
array
Special matrix creation
zeros(M,N)
&gt;&gt; A=zeros(2,3)
A=
Matrix of zeros
ones(M,N)
&gt;&gt; B=ones(2,2)
B=
Matrix of ones
eye(M,N)
&gt;&gt; C=eye(2,2)
C=
Matrix of ones on the
diagonal
rand(M,N)
Matrix of random
numbers between 0
and 1
&gt;&gt; D=rand(3,2)
D=
0
0
0
0
0
0
1
1
1
1
1
0
0
1
0.9501
0.4860
0.2311
0.8913
0.6068
0.7621
Operations on vectors and matrices in Matlab
Math
Matlab
A+B
A-B
Multiplication/ division
(element by element)
A.*B
A./B
Multiplication
(Matrix Algebra)
A*B
Transpose:
AT
A’
Inverse:
A-1
inv(A)
Determinant:
|A|
det(A)
“single quote”
Array Operations
Scalar-Array Mathematics
Addition, subtraction, multiplication, and division of an array by a scalar
simply apply the operation to all elements of the array.
&gt;&gt; f = [1 2 3; 4 5 6]
f =
1 2 3
4 5 6
&gt;&gt; g = 2*f -1
g =
1 3 5
7 9 11
Array Operations
Element-by-Element Array-Array Mathematics
When two arrays have the same dimensions, addition, subtraction,
multiplication, and division apply on an element-by-element basis.
Operation
Subtraction
Multiplication
Division
Exponentiation
Algebraic Form
a+b
a−b
axb
a/b
ab
Matlab
a + b
a - b
a.*b
a./b
a.^b
Array Operations
 a11 a12

A   a 21 a 22
a
 31 a 32
a13 

a 23 
a 33 
M
 b11 b12

B   b 21 b 22
b
 31 b 32
b13 

b 23 
b33 
N
N
 a11  b11 a12  b12

A  B   a21  b21 a22  b22
a b
 31 31 a32  b32
N
a13  b13 

a23  b23 
a33  b33 
M
M
Array Operations
 1 2 3


B   4 5 6
 7 8 9


 1 2 3


A   4 5 6
 7 8 9





AB



2
4
6
8
10
12
14
16
18










AB



0
0
0
0
0
0
0
0
0







Array Operations
Element-by-Element Array-Array Mathematics
&gt;&gt; A
&gt;&gt; B
&gt;&gt; C
C =
4 15
= [2 5 6];
= [2 3 5];
= A.*B
30
&gt;&gt; D = A./B
D =
1.0000 1.6667 1.2000
&gt;&gt; E = A.^B
E =
4 125 7776
&gt;&gt; F = 3.0.^A
F =
9 243 729
Array Operations
MATRIX Multiplication (element by element)
 a11 a12

A   a 21 a 22
a
 31 a 32
a13 

a 23 
a 33 
M
 b11 b12

B   b 21 b 22
b
 31 b 32
b13 

b 23 
b33 
N
N
NOTATION
“dot”
“multiply”
 a11  b11 a12  b12

A  B   a21  b21 a22  b22
 a b
 31 31 a32  b32
N
a13  b13 

a23  b23  M
a33  b33 
M
Array Operations
Examples: Multiplication &amp; Division
(element by element)
 1 2 3


B   4 5 6
 7 8 9


 1 2 3


A   4 5 6
 7 8 9





A  B  



1
4
9
16
25
36
49
64
81






A

/
B









1
1
1
1
1
1
1
1
1







Array Operations
Matrix Multiplication
The matrix multiplication of m x n matrix A and nxp matrix B yields
m x p matrix C, denoted by
C = AB
Element cij is the inner product of row i of A and column j of B
n
cij   aik bkj
k 1
Note that
AB ≠ BA
Array Operations
 a11 a12

A   a 21 a 22
a
 31 a 32
a13  Row 1

a 23  M1
a 33 
Column 1
Matrix Multiplication
 b11 b12

B   b 21 b 22
b
 31 b 32
N1
b13 

b 23  M2
b33 
N2
N1=M2
NOTATION
 a11b11  a12b 21  a13b31 a11b12  a12b 22  a13b32

A  B   a 21b11  a 22b 21  a 23b31 a 21b12  a 22b 22  a 21b32
a b  a b  a b
 31 11 32 21 33 31 a 31b12  a 32b 22  a 33b32
N2
“multiply”
a11b13  a12b 23  a13b33 

a 21b13  a 22b 23  a 23b33 M1
a 31b13  a 32b 23  a 33b33 
Array Operations
Example: Matrix Multiplication
1 2 3


A  3 2 1
 3 1 2





AB  



1 2 3


B  3 2 1
 3 1 2


1x1 + 2x3 +3x3
1x2 + 2x2 +3x1
1x3 + 2x1 +3x2
16
9
11
12
11
13
12
10
14







Array Operations
Solving systems of linear equations
Example: 3 equations and 3 unknown
1x + 6y + 7z =0
2x + 5y + 8z =1
3x + 4y + 5z =2
Can be easily solved by hand, but what can we do if it we have 10 or 100
equations?
Array Operations
Solving systems of linear equations
1x + 6y + 7z = 0
2x + 5y + 8z = 1
3x + 4y + 5z = 2
First, write a matrix with all the (xyz)
coefficients
Write a matrix with all the constants
 0
 
B  1
 2
 
1 6 7


A  2 5 8
 3 4 5


Finally, consider the matrix of unknowns
x
 
S   y
z
 
Array Operations
Solving systems of linear equations
AxS=B
A-1 x A x S = A-1 x B
(A-1 x A) x S = A-1 x B
I x S = A-1 x B
S = A-1 x B
Array Operations
Solving systems of linear equations
The previous set of equations can be expressed in the following vector-matrix
form:
AxS=B
1x + 6y + 7z =0
2x + 5y + 8z =1
3x + 4y + 5z =2
1 6 7  x   0

    
 2 5 8 X  y  1
 3 4 5  z   2

    
Array Operations
Matrix Determinant
•The determinant of a square matrix is a very useful value for finding if a system
of equations has a solution or not.
•If it is equal to zero, there is no solution.
Notation:
Determinant of A = |A| or det(A)
Formula for a 2x2 matrix:
 m11 m12 

M  
 m 21 m 22 
det(M)= m11 m22 – m21 m12
IMPORTANT: the determinant of a matrix is a scalar
Array Operations
Matrix Inverse
•The inverse of a matrix is really important concept, for matrix algebra
•Calculating a matrix inverse is very tedious for matrices bigger than 2x2.
We will do that numerically with Matlab.
Notation:
inverse of A = A-1 or inv(A)
Formula for a 2x2 matrix:
 m11 m12 

M  
 m 21 m 22 
M-1=
1  m22  m12 


det( M )   m21 m11 
IMPORTANT: the inverse of a matrix is a matrix
Array Operations
Matrices properties
Property of inverse :
A x A-1 = I
and
A-1 x A = I
Example:
0.6   1 0 0 
 1 1 2    0.4 0.2

 
 

 1 2 1 x  0.2 0.6  0.2    0 1 0 
 2 0 1   0.8  0.4  0.2   0 0 1 

 
 

Property of identity matrix:
and
IxA=A
AxI=A
Solving systems of equations in Matlab
x + 6y + 7z =0
2x + 5y + 8z =1
3x + 4y + 5z =2
In Matlab:
&gt;&gt; A=[ 1 6 7; 2 5 8; 3 4 5]
&gt;&gt; B=[0;1;2];
&gt;&gt; S=inv(A)*B
1 6 7
x


 
A  2 5 8 S   y
 3 4 5
z


 
Verification:
&gt;&gt; det(A)
ans =
&gt;&gt;
S=
0.8571
-0.1429
0
28
 0
 
B  1
 2
 
Solving systems of equations in Matlab
x + 6y + 7z =0
2x + 5y + 8z =1
3x + 4y + 9z =2
In Matlab:
&gt;&gt; A=[ 1 6 7; 2 5 8; 3 4 5]
&gt;&gt; B=[0;1;2];
&gt;&gt; S=inv(A)*B
1 6 7
x


 
A  2 5 8 S   y
3 4 9
z


 
 0
 
B  1
 2
 
Verification:
&gt;&gt; det(A)
ans =
Warning: Matrix is singular to working precision.
&gt;&gt;
S=
NaN
NaN
NaN
0
NO Solution!!!!!
Applications in mechanical engineering
y
F1
7N
20o
30o
5N
60o
80o
F2
Find the value of the forces F1and F2
x
Applications in mechanical engineering
y
F1
7N
20o
30o
5N
Projections on the X axis
60o
x
80o
F2
F1 cos(60) + F2 cos(80) – 7 cos(20) – 5 cos(30) = 0
Applications in mechanical engineering
y
F1
7N
20o
30o
5N
Projections on the Y axis
60o
x
80o
F2
F1 sin(60) - F2 sin(80) + 7 sin(20) – 5 sin(30) = 0
Applications in mechanical engineering
F1 cos(60) + F2 cos(80) – 7 cos(20) – 5 cos(30) = 0
F1 sin(60) - F2 sin(80) + 7 sin(20) – 5 sin(30)
=0
F1 cos(60) + F2 cos(80)
F1 sin(60) - F2 sin(80)
= 7 cos(20) + 5 cos(30)
= - 7 sin(20) + 5 sin(30)
In Matlab, sin and cos use radians, not degree
In Matlab:
&gt;&gt; CF=pi/180;
&gt;&gt; A=[cos(60*CF), cos(80*CF) ; sin(60*CF), –sin(80*CF)];
&gt;&gt; B=[7*cos(20*CF)+5*cos(30*CF) ; -7*sin(20*CF)+5*sin(30*CF) ]
&gt;&gt; F= inv(A)*B or (A\B)
Solution:
F=
F1= 16.7406 N
16.7406
F2= 14.6139 N
14.6139
```