2015 – Semester 2

Lecturer: Dr Tony Vo

Slides by: Dr Tony Vo

Photo credit: Janet Waters

Lecture 3

Matrices

ENG1060: Computing for Engineers

Using MATLAB as a calculator (command window)– A = 3 * 10 ^ 4 + 5 / 8– A = 3 * 10 ^ (4 + 5) / 8 (order of operation)

Proper variable naming– Queensland | Queen | Que | Q

Built-in functions– sin, cos, log, floor, round

Solving problems– Think before you do– Challenge your answers (realistic?)– Do your work in m-files

Lecture 2 recapENG1060: Computing for Engineers

What are matrices?

Creating matrices

Matrix addressing

Lecture 3 overviewENG1060: Computing for Engineers

A matrix is a variable with multiple values– Matrices allow us to perform many calculations– Matrices are often called arrays in programming

MATLAB stands for MATrix LABoratory– MATLAB excels at matrix operations

Matrices are used everywhere in engineering and science

What is a matrix?ENG1060: Computing for Engineers

Matrix structure: rows-by-columns

Multi-dimensional matrices

𝐴𝐴 = 1 (scalar)

𝐵𝐵 = 1 2 3 4 5 (row vector, 1-by-5 matrix)

C=

12345

(column vector, 5-by-1 matrix)

D=147

258

369

47

10(3-by-4 matrix)

ENG1060: Computing for Engineers

Thinking about it physically…– 0D matrices are points– 1D matrices are edges/sides– 2D matrices are faces– 3D matrices are surfaces

Multi-dimensional matrices

An edge A face A surface

ENG1060: Computing for Engineers

Audio example– playaudio.m– .wav: Less than 1 second of audio and it's a 13680 × 1 matrix– .mp3: A 597883 × 2 matrix

Graphics example– playimage.m– Black and white image: 135 × 90– Colour image: 135 × 90 × 3

Other examplesENG1060: Computing for Engineers

Matrices are used everywhere in engineering and science– We deal with large and complex problems

Audio processing– E.g. 2 minute audio contains 5,777,100 data

Image processing– E.g. 10 Megapixels colour image contains 2592 × 3872 × 3 =

30,108,672 data

Video processing– E.g. High speed video (1280 x 1024, 1000 fps) contains 1,310,720,000

data per second

Why use matrices?ENG1060: Computing for Engineers

Humans are incapable of retaining a lot of information and we're slow!– Instead we use computers to store data in matrices

Example: Hawkeye (Tennis video referee)– 10 cameras, each capturing 1000 images/second– Each image is 1280 x 1024 pixels (1.3 Megapixels)– In 1 second, 1.2 × 1010 values need to be processed

Matrix exampleENG1060: Computing for Engineers

Square brackets [ ] to create matrices

Row vectors are horizontalSyntax: A = [1 2 3 4] or A = [1, 2, 3, 4]

Column vectors are verticalSyntax: B = [5; 6; 7; 8]

To convert row vectors to column vectors and vice versa– Use the transpose operator ' (apostrophe)– Or use the transpose command

Creating 1D matrices (vectors)ENG1060: Computing for Engineers

What if I require numbers from 1 to 106?

We do not want to type it all out! – Remember, engineers are lazy…

Typing it outENG1060: Computing for Engineers

Colon operator :– Creates a vector with equally spaced values using a specified spacing

Syntax: start_value : step : end_value– Default step of 1 if step is omitted

Examples:A = [6 7 8 9 10] or A = 6 : 1 : 10 or A = 6 : 10B = [6 11 16 21 26] or B = 6 : 5 : 26C = [6 4 2 0 -2 -4] or C = 6 : -2 : -4

The colon operatorENG1060: Computing for Engineers

How to create a vector with the following requirements?– First value is 3– Last value is less than or equal to 40– Step size of 5

What if the value of 40 is needed?– Is the end value important?

Colon operator to create column vectors?– Use round brackets with apostrophe: A = (1 : 20)'

The colon operator: LimitationsENG1060: Computing for Engineers

What if I require 7 equally spaced numbers between 3.851 and 7.84?– I can manually calculate the step size

Step size = (end value – start value)/(number of points - 1)

Step size = 7.84 −3.8517 −1

= 0.6648

→ A = 3.851 : 0.6648 : 7.84

Is there an alternative?

What if …ENG1060: Computing for Engineers

The linspace function:– Create a vector with equally spaced values using a specified number of

points

Syntax: linspace(start_value, end_value, num_of_points)– Default of 100 points if num_of_points is omitted

Examples:A = [1 2 3 4 5 6] or linspace(1, 6, 6)

B = [3.8510 4.5158 5.1807 5.8455 6.5103 7.1752 7.84]or B = linspace(3.851, 7.84, 7)

The linspace functionENG1060: Computing for Engineers

You are working as a consultant for a company that installs street lamps

Objective: They want these lamps to be spread out in 20 metre intervals along the street

You have been asked to determine the displacements (x) to install lamps along a street that is 400 metre long

MATLAB command:X = ???

Example: Use of the colon operatorENG1060: Computing for Engineers

They've changed their minds. They now want a fixed number of lamps to be spread evenly along the street using 23 lamps

You have been asked to determine the displacements (x) to install 23 lamps along a street that is 525 metres long

MATLAB command:X = ???

Example: Use of the linspace functionENG1060: Computing for Engineers

Both create vectors of equally spaced values– Colon operator uses a specified step size– Linspace function uses a specified number of values

Use both appropriately throughout your coding in this unit

What does logspace do?

Colon operator vs. linspace functionENG1060: Computing for Engineers

A 2-dimensional matrix contains multiple rows and columns

Use square brackets [ ] to create a 2 dimensional matrix– MATLAB refers to the rows first, then the columns

A = [1 2 3; 4 5 6] (2 × 3 matrix)

Use the colon operator to create vectors within matricesA = [1:3; 4:6]

Creating 2D matricesENG1060: Computing for Engineers

Matrices can be concatenated to make larger matrices– Matrices can only join if one of their dimensions are the same– Dimension mismatch in MATLAB will give an error in red text

Matrix concatenation

arms =

7777

7777

body =

1001

0101

0011

dino = 9 9 99 9 9

9 99 9

9 99 9

Possible to concatenate two arms a body and a dino?X = [arms; body dino] ?X = [arms; body; dino] ?X = [arms body arms; dino] ?

ENG1060: Computing for Engineers

X = [arms body arms; dino]

Matrix concatenation: Result

X =

7 7 1 0 0 7 77 7 0 1 0 7 77 7 0 0 1 7 77 7 1 1 1 7 79 9 9 9 9 9 99 9 9 9 9 9 9

ENG1060: Computing for Engineers

Similar to the colon operator and linspace

Matrices can be created using:A = zeros(rows, columns)B = ones(rows, columns)C = eye(rows, columns)D = rand(rows, columns)

Matrix properties can be obtained using Longest_side = length(matrix)[rows, columns] = size(matrix)

Built-in functions for matricesENG1060: Computing for Engineers

We have already seen that matrices can contain a lot of information– How do we display the data? Is the entire matrix needed?

What would Y = rand(1000, 1000) print to screen?– Not important to show but important for future calculations

To suppress an output in MATLAB, place a semi-colon (;) after the command

Y = rand(1000, 1000);

Information overloadENG1060: Computing for Engineers

The semi-colon is used to suppress the printing of outputs– Recall: semi-colon creates a new row in the matrix environment [ ]

Example: We're only interested in the final resultA = 10;B = 40;C = 2*A – B + 887/B;D = A*B*C – B + 887/C

The semi-colonENG1060: Computing for Engineers

All values in a vector or matrix are assigned an address– Refer to numbers within a vector or a matrix to perform calculations– Create smaller matrices from a larger matrix

To address elements in a vectorSyntax: A(index)

Example: A = [5 10 15 20 25 30 35 40]A(1) → 5 (first element)A(4) → 20 (fourth element)A(end) → ???A(first) → ???

Matrix addressing: VectorsENG1060: Computing for Engineers

So we can use a scalar index to address an element vector

What if we want multiple values from a vector? A = [10, 20, 30, 40, 50, 60]Individually… A(1) = 10, A(4) = 40, A(5) = 50 and A(6) = 60

Alternatively, we can use an index that is a vectorA([1 4 5 6]) = [10, 40, 50, 60] any other ways?

Matrix addressing: VectorsENG1060: Computing for Engineers

Remember that– Square brackets [ ] are used to create matrices– Round brackets ( ) are used to address matrices

Example:A = [6, 9, 15, 20, 98, 241, 259]A([1, 3:5, 7]) = [6, 15, 20, 98, 259]

If elements of a vector have an index (or an address), do elements of other multi-dimensional matrices have an index?

Matrix addressing: SyntaxENG1060: Computing for Engineers

A 2D matrix has rows and columns– Therefore we give an index to the element's row and column

MATLAB takes in the row argument first, then the columnSyntax: A(row_index, column_index)

Again, vectors can be used as indices

A colon (:) by itself tells MATLAB to return either all rows or all columns

Matrix addressing: 2D matricesENG1060: Computing for Engineers

Matrix addressing: 2D matricesENG1060: Computing for Engineers

Consider the following matrixT = [7 1 5 9; 2 6 4 9; 8 9 3 9]

T =7 1 5 92 6 4 96 9 3 9

Matrix addressing: Example

What would the following commands return?D = T( :, [2 4])Z = T(end-1, [1 3 4])B = [Z(1) Z(end)]

ENG1060: Computing for Engineers

1. Introduction to ENG10602. MATLAB basics3. Matrices4. Matrix calculations and plotting5. Good programming practices6. Functions7. Input and output8. IF statements9. Loops, loops, loops…10. Debugging MATLAB programs11. Advanced functions and limitations of MATLAB12. The workings of MATLAB

Part A: MATLAB programmingENG1060: Computing for Engineers