Learner’s Guide to MATLAB®

Chapter 2 : Working with Arrays

Working with Arrays

Chapter Outline

• Creating Vector and Matrix

• Accessing and Addressing Matrix

• Mathematical Operations with Matrix

• Functions for Analyzing Matrix

• Chapter Case Study

Working with Arrays

Creating Vectors

a

n subdivision points

bdx

>> x = a:dx:b;

>> x = linspace(a,b,n);

>> x = x';

x

row vectors

convert to column vector

Working with Arrays

Creating Vectors - Examples

1) Create a COLUMN VECTOR with the elements :55, 14, log(51), 0, sin(pi/3)

2) Create a ROW VECTOR which the first element is 1, the last element is 33, with a increment of 2 between elements

Working with Arrays

Creating Matrices

>> A = [1,2,3; 4,5,6; 7,8,9];

>> A = [1 2 3; 4 5 6; 7 8 9];

>> A = [1 2 34 5 67 8 9]

>> A = A'

or

or

data entry mode

transpose

Working with Arrays

Array Operations

A = 1 2 3 4 5 6 7 8 9

B = 1 2 4 3 4 5 6 5 4

C = A + B

C =

2 4 7 7 9 11 13 13 13

A = 1 2 3 4 5 6 7 8 9

B = 1 2 4 3 4 5 6 5 4

C = A.*B

C =

1 4 12 12 20 30 42 40 36

Array addition Array multiplication

A = 1 2 3 4 5 6 7 8 9

C = A.^2

C =

1 4 9 16 25 36 49 64 81

Array power

Working with Arrays

Array Multiplication

• Matrices must have the same dimensions

• Dimensions of resulting matrix = dimensions of multiplied matrices

• Resulting elements = product of corresponding elements from the original matrices

>> a = [1 2 3 4; 5 6 7 8];

>> b = [1:4; 1:4];

>> c = a.*b

c =

1 4 9 16 5 12 21 32

>> a = [1 2 3 4; 5 6 7 8];

>> b = [1:4; 1:4];

>> c = a.*b

c =

1 4 9 16 5 12 21 32 c(2,4) = a(2,4)*b(2,4)

Working with Arrays

Matrix Operations

A = 1 2 3 4 5 6 7 8 9

B = 2*A

B =

2 4 6 8 10 12 14 16 18

A = 1 2 3 4 5 6 7 8 9

B = 2 + A

B =

3 4 5 6 7 8 9 10 11

Scalar Multiplication Scalar Expansion

A = 1 2 3 4 5 6 7 8 9

B = 1 2 4 3 4 5 6 5 4

C = A * B

C =

25 25 26 55 58 65 85 91 104

Matrix Multiplication

Working with Arrays

>> e=[1 2;3 4] + 5

1 2 = + 5 3 4

1 2 5 5 = + 3 4 5 5

6 7= 8 9

>> e=[1 2;3 4] + 5

1 2 = + 5 3 4

1 2 5 5 = + 3 4 5 5

6 7= 8 9

Matrix Calculation-Scalar Expansion

>> e=[1 2;3 4] + 5e = 6 7 8 9

>> e=[1 2;3 4] + 5e = 6 7 8 9

Scalar expansion

Working with Arrays

Matrix Multiplication

• Inner dimensions must be equal.

• Dimension of resulting matrix = outermost dimensions of multiplied matrices.

• Resulting elements = dot product of the rows of the 1st matrix with the columns of the 2nd matrix.

>> a = [1 2 3;4 5 6];

>> b = [3,1;2,4;-1,2];

>> c = a*b

c =

4 15 16 36

>> a = [1 2 3;4 5 6];

>> b = [3,1;2,4;-1,2];

>> c = a*b

c =

4 15 16 36

[2x3]

[3x2]

[2x3]*[3x2] [2x2]

a(2nd row).b(2nd column)

Working with Arrays

Array Addressing

m(2:4,3)

m(3,1)

Working with Arrays

More Example on indexing

>> a=[3 11 6; 4 7 10; 13 9 0]

a =

3 11 6

4 7 10

13 9 0>> a(3,1)= 20a = 3 11 6 4 7 10 20 9 0>> a(2,3)-a(1,2)ans = -1

>> a=[3 11 6; 4 7 10; 13 9 0]

a =

3 11 6

4 7 10

13 9 0>> a(3,1)= 20a = 3 11 6 4 7 10 20 9 0>> a(2,3)-a(1,2)ans = -1

Create a 3 x 3 matrixUse square brackets [ ]

Matrices must be rectangular. (Undefined elements set to zero)

Assign a new value to the (3,1) element

Use elements in a mathematical expression

Working with Arrays

More Example on Colon indexing

>> a=[3 11 6; 4 7 10; 13 9 0]

a =

3 11 6

4 7 10

13 9 0>> b = a(:,3)b = 6 10 0 >> c = a(2,:)c =

4 7 10>> d = a(2:3,1:2)d = [4 7 ] [13 9 ]

>> a=[3 11 6; 4 7 10; 13 9 0]

a =

3 11 6

4 7 10

13 9 0>> b = a(:,3)b = 6 10 0 >> c = a(2,:)c =

4 7 10>> d = a(2:3,1:2)d = [4 7 ] [13 9 ]

Create a 3 x 3 matrix

Define a column vector b from elements in all rows of column 3 in matrix a

Define a row vector c from elements in all columns of row 2 in matrix a

Create a matrix d from elements in rows 2&3 and columns 1&2 in matrix a

Working with Arrays

• Solve this set of simultaneous equations

Array Division using “Left Division”

>> A = [-1 1 2; 3 -1 1;-1 3 4];

>> b = [2;6;4];

>> x = inv(A)*b

x =

1.0000

-1.0000

2.0000

>> x = A\b

x =

1.0000

-1.0000

2.0000

>> A = [-1 1 2; 3 -1 1;-1 3 4];

>> b = [2;6;4];

>> x = inv(A)*b

x =

1.0000

-1.0000

2.0000

>> x = A\b

x =

1.0000

-1.0000

2.0000

-x1 + x2 + 2x3 = 2

3x1 - x2 + x3 = 6

-x1 + 3x2 + 4x3 = 4

Working with Arrays

Function Description Example

C=max(A) If A is vector, C is the largest element in A

A = [5 9 2]

C = max(A)

sum(A) If A is vector, returns the sum of elements of A

A = [5 9 2]

sum(A)

sort(A) If A is vector, arranges elements of vector in ascending order

A = [5 9 2]

sort(A)

det(A) Returns the determinant of a square matrix A

A = [2 4; 3 5];

det (A)

inv(A) Returns the inverse of a square matrix A

A = [2 4; 3 5];

inv(A)

Functions for Analyzing Matrix

Working with Arrays

Sample Problem 2 : Friction Experiment

The coefficient of friction, μ, can be determined in an experiment by measuring the force F required to move a mass m. When F is measured and m is known, the coefficient of friction can be calculated by:

µ = F / (mg) where g = 9.81 m/s2

Results from measuring F in six tests are given in the table below. Determine the coefficient of friction in each test, and the average from all tests.

Test # 1 2 3 4 5 6

Mass m (kg) 2 4 5 10 20 50

Force F (N) 12.5 23.5 30 61 117 294

Working with Arrays

Summary

• Creating Vector and Matrix

• Accessing and Addressing Matrix

• Mathematical Operations with Matrix

• Functions for Analyzing Matrix