matrices and matlab dr viktor fedun automatic control and systems engineering, c09 based on lectures...

Matrices and MATLAB

Dr Viktor FedunAutomatic Control and Systems Engineering, C09

Based on lectures by Dr Anthony Rossiter

Examples of a matrix


A matrix can be thought of simply as a table of numbers with a given number of rows and columns.

Example A has 3 rows and 3 columns, example B has 4 rows and 4 columns.

212354

8687

5254

6432

;

413

965

302

BAExamples of

square matrices


A matrix can be thought of simply as a table of numbers with a given number of rows and columns.

Example A has 2 rows and 3 columns, example B has 3 rows and 1 column.

7

6

5

;501

432BA

Examples of non-square

matrices

What use are matrices?• Compact form for storing data (equivalent to

a table but more flexible).• Each column(or row) can relate to a different

‘state’: e.g. Time, displacement, velocity, temperature, etc.

• Convenient for computer code (and algebra) as can store and share large quantities of related data with a single variable.

• Easily extend to dynamic relations/modelling.

Indexing of matrices

Always give as {row index,column index}

Start counting from top left, so the top left component is index {1,1}, e.g. indices are given as:

}4,4{}3,4{}2,4{}1,4{

}4,3{}3,3{}2,3{}1,3{

}4,2{}3,2{}2,2{}1,2{

}4,1{}3,1{}2,1{}1,1{

Row index increasing

Column index

increasing

What is the index of the component -4

212354

8687

5254

6432

B

1. {1,3}

2. {1,2}

3. {2,1}

4. {3,1}

5. {4,3}

What is the index of the component -4

212354

8687

5254

6432

B

1. {1,3}

2. {1,2}

3. {2,1}

4. {3,1}

5. {4,3}

}4,4{}3,4{}2,4{}1,4{

}4,3{}3,3{}2,3{}1,3{

}4,2{}3,2{}2,2{}1,2{

}4,1{}3,1{}2,1{}1,1{

Create a matrix with following information

2nd row and 3rd column is 5

4th row and 2nd column is 2

1st row and 5th column is 6

Remainder are 0.

TRY this on MATLAB

A(2,3)=5; A(4,2)=2; A(1,5)=6;disp(A)

Note MATLAB syntax

matches that used in

mathematics, i.e. row index

and then column index.

Extracting parts of matrices

Example of selecting given rows and columns.

Can rearrange the order if desired.

Matrices

The main thing to note is that the default variable in MATLAB is a matrix (or vector if the row or column dimension is one).

Any BODMAS type operation that is valid with matrices can be carried out with the same syntax.

MATLAB also includes a large number of matrix analysis tools that you will find useful in the 2nd year.

Matrices and MATLAB for plots

A student collects data from an experiment and stores the answer in a matrix data_exp1.

• Column 1 corresponds to time.• Column 2 to temperature• Column 3 is voltage supplied.

It is now simple to code and plot this data – the corresponding code is shown next.

0 20 40 60 80 100 1200

5

10

15

20

25

30

Time (sec)

Temperature

Voltage

MATLAB example

Correcting dataStudent is told that the temperature readings

were all out by 2 degrees due to a calibration error.

Once the data is in a matrix, update is very efficient.

Use line: data_exp1(:,2)=data_exp1(:,2)+2

This will work not matter how many items of data are affected.

Matrix notationWithin a textbook, it is more normal to use

subscripts to represent the indices, so for example:

4,43,42,41,4

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

AAAA

AAAA

AAAA

AAAA

A

Often the comma is omitted for low value integers

131211 BBBB

Matrix notation

Flexibility

As an engineer you need to be flexible and work out what notation a particular article, website, book, etc is using.

Usually it will be obvious from the context and usage as long as your understand the core principles.

Which statement is false?

042

056

238

D

1. D12=-3

2. D21=6

3. D32=2

4. D23=2

5. D33=0

Matrix dimensionsWhat are the dimensions of the following

matrices?

3

5

4

042

154

233

;

042

056

238

ED

7

6

5

;501

432BA

0404512 C

Matrix dimensions summary

• Dimensions are always number of rows first and then number of columns second.

Typical notation is 2 by 3 or 4 by 1 or similar.

Describe dimensions of matrices on previous slide.

• Indexing begins from top left corner.

MATRIX TRANSPOSE

This is a very common operation so it is critical that students are familiar with the notation.

There is nothing to understand as it is simply a definition.

Matrix transpose definitionThe transpose is determined by swapping

the position of all the indices, that is {1,2} becomes {2,1}, etc.

You could view this a bit like doing a reflection about the principle diagonal.

Diagonal elements do not change.

;43

32;

43

32

TAA

Tijji AA ,,

Examples of Matrix transposeTijji AA ,,

443

362;

4

4

3

632

TAA

0

4

04

5

21

;0404512 TCC

Alternative insight.The jth column of A is the jth

row of A transpose.

Matrix transpose examplesFind the transpose of the following:

3

5

4

042

154

233

042

056

238

E

D

7

6

5

B

Which is the correct transpose of G?

1. A

2. B

3. C

4. D

5. None of these

2

2

2

2

2

2;

2

2

2

2

2

2

2

2

2

2

2

2;

2

2

2

2

2

2;

2

2

2

222

DC

BAG

ADDITION AND SUBTRACTION OF MATRICES

Addition and subtraction of matrices

Add (or subtract) components in same position, that is with the same {row, column} index.

Matrices must be same dimensions or you cannot add them!

24

53;

67

21;

43

32BABA

jijiji BACBAC ,,,

Do following matrices computations

BABA ;67

24;

43

32

BABA 2;

7

6

8

1202

;4

4

3

632

Which is incorrect (could be several)?

9

4

8

2

2

0;

91

43

1

2

2

0

2

2;

52

31

;8

2

6

2

0

2;

43

12

EF

DC

AG

1. G+C=F

2. G-C=-F

3. A+D=E

4. D-A=-E

5. None of these

Do following matrix computations

BABA T 3;67

24;

43

32

TBABA ;

4

6

0

102

;0

5

3

132

MATRIX handling in MATLAB

• Matrix addition/subtraction

• Matrix multiplication• Matrix powers• Matrix inversion• Extracting parts of

matrices

• A+B, A-B, A-2*B+C• A*B• A^3 • inv(A) or A^(-1)

• A([1,3],[2, 4])

MATRIX MULTIPLICATION

This is the most common skill you will need.

The technique is simply a definition that you need to memorise.

There is no such thing as matrix division!

Basic rule for multiplication (BY DEFINITION – not to understand)

Index {i,j} of result is determined from row {i} of left hand matrix and column {j} of right hand matrix.

Multiplying a row on left by a column on the right is a ‘dot-product’ type of operation, e.g.

dhcgbfae

h

g

f

e

dcba

What is the result of the following ?

8

6

2

533

1. 88

2. 52

3. 32


2

04

3

2

72113

1. 18

2. 21

3. 27

4. Don’t know


3

0

1

21

1. 1

2. 6

3. NA

4. Not sure

KEY OBSERVATIONA row vector can only be multiplied onto a

column vector if the two vectors are the same length.

Number of columns of left hand vector must match the number of rows of the right hand vector.

If vectors have different dimensions, multiplication is not defined.

Rules for matrix multiplicationAssume we wish to do C=A*B.

A must have same number of columns as B has rows

C BARows of C

Columns of B

Rows of B

Columns of C

Columns of A

Rows of A

=

Basic rule for multiplication Index {i,j} or result is determined from row {i} of left hand matrix and column {j} of right hand matrix. Multiplying a row on left by a column on the right is a ‘dot-product’ type of operation, e.g.

phmgnfke

dhcgbfae

h

g

f

e

p

d

m

c

n

b

k

a

fpej

dpcj

bpaj

fmei

dmci

bmai

fneh

dnch

bnah

fkeg

dkcg

bkag

p

j

m

i

n

h

k

g

f

d

b

e

c

a

Student problemSolve the following multiplications.

21226

11054

1971

8

6

6

5

4

2

26

54

71

8

6

6

5

4

2

Student problem 2 – compute C,D

BADABCBA

;;

26

54

71

;68

46

6

5

4

2

KEY OBSERVATIONTwo matrices can only be multiplied if the

column dimension of the left matrix matches the row dimension of the right matrix.

In general

In fact, it is possible that AB exists but BA does not!

BAAB The property of

AB=BA is called commutativity and does not

hold for matrices in general.

Uses of matrix multiplicationA simple example involves the change of coordinates from one set to another – this is commonly needed for robotics and mechanics.

x

y

X’Y’

θ

V=(x,y)

cossin'

sincos'

yxY

yxX

y

x

Y

X

cossin

sincos

'

'

SIMULTANEOUS EQUATIONS

Matrix/vector algebra is a compact and efficient method for handling linear simultaneous equations and most effective techniques deployed assume matrix representation of this problem. [Details in semester 2]

MATLAB automatically handles simultaneous equations with matrices and vectors which greatly simplifies solution and manipulation. [Try this now!]

Using Matrices for simultaneous equations

Consider a single linear equation and write in MATRIX/VECTOR format, e.g.

6]61[65

4]23[423

y

xyx

y

xyx

6

4

51

23

5

23

y

x

yx

yx

Put following into matrix-vector format

1041320

463

32

23

165

zyx

yz

yzx

yx

yx

91412

214

432

12

dab

dca

dcb

cba

Simultaneous equationsAssume user knows how to put simultaneous

equations into matrix vector format, ie.

AX=b

Can solve with MATLAB several different ways;

1. X=inv(A)*b

2. X=A\b

3. Numerically efficient code is discussed in text books and semester 2 of ACS123.

Try on MATLAB

820

241

012

;

2

3

1

;1 AbbAx

DETERMINANTS

You will cover solution of simultaneous equations in semester 2. However, in order to prepare for this, you will need familiarity and skill with the determinant of a matrix.

Final matrix topic here is define ‘determinant’.

NOTE: This is a definition and therefore must be learnt as a fact.

ONLY DEFINED FOR SQUARE MATRICES

Determinant in briefIn conceptual terms, a determinant gives an indication of the ‘magnitude’ of a matrix, that is, on average, what scale of impact does it have on vectors when multiplying those vectors.

You will notice that the effective magnification is direction dependent, and so the determinant is an average in some sense.

;1

1

1

1

32

23;

5

5

1

1

32

23

Scaled by factor of 5. Scaled by factor of 1.

Determinant – key factIf the determinant is very small, then there exists a least one direction, which when multiplied by the matrix, results in a very small scaling factor.

A zero determinant means at least one direction can be mapped to zero.

;0

1.0

9.0

1.0

218

110;

20

11

1

1

218

110

Scaled by factor of 16 (approx).

Scaled by factor of 0.1 (approx).

Determinant for 2 by 2 matricesThe determinant is defined as product of the diagonal elements minus the product of the off-diagonal elements:

Note use of vertical lines around matrix is notation use to define determinant.

Try this on the examples of the previous few slides.

bcadAdc

baA

;

Determinant computations

AA ;

32

23

BB ;

218

110

Find the determinant of matrix D

85

41D

1. -12

2. 28

3. 12

4. -28

5. unsure

Find the determinant of matrix F

124

31F

1. 0

2. 24

3. -24

4. -32

5. unsure

Determinant for 3 by 3 matricesThe determinant is defined using the concept of a matrix of cofactors Aij. For convenience define the original matrix with lower case and the matrix of cofactors with upper case.

Next, by definition:

333231

232221

131211

333231

232221

131211

)(;

AAA

AAA

AAA

Acof

aaa

aaa

aaa

A

333332323131

232322222121

131312121111

AaAaAaor

AaAaAaor

AaAaAaA

Defining the cofactor termsA cofactor is taken as the determinant of a minor with a sign change if appropriate to the position.

A minor is the matrix remaining when the row and column associated to an element are crossed out.

3231

222113

333231

232221

131211

)(min;aa

aaaor

aaa

aaa

aaa

A

;

sign 312232211313 )( aaaaacofA

Defining the cofactor termsA cofactor is taken as the determinant of a minor with a sign change if appropriate to the position.


3332

131221

333231

232221

131211

)(min;aa

aaaor

aaa

aaa

aaa

A

;

sign ][)( 133233122121 aaaaacofA

Find the matrix of cofactors and determinant

210

018

563

A131312121111 AaAaAaA

Find the matrix of cofactors and determinant

512

212

430

A

Determinant for 4 by 4 matricesThe determinant is defined using the concept of a matrix of cofactors Aij (as for the 3 by 3 case)

For convenience define the original matrix with lower case and the matrix of cofactors with upper case.

Next, by definition:

4

1

1414131312121111

jijijAaor

AaAaAaAaA

The extension to larger matrices should be obvious, although this is not a paper and pen exercise.

Defining the cofactor terms for a 4 by 4

A cofactor is taken as the determinant of a minor with a sign change if appropriate to the position.


4,43,42,41,4

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

aaaa

aaaa

aaaa

aaaa

A

Remarks on 4 by 4 determinantsA minor is now a 3 by 3 matrix and therefore

each cofactor requires the computation of the determinant of a 3 by 3

For a 5 by 5 matrix, the minors would each require the computation of a 4 by 4 determinant.

Determinants of larger matrices are best done using properties and tricks to minimise the numeric

computations. Alternatively, and more likely, use a computer.

Properties of determinants1. Adding a multiple of any row to another row does not

change the determinant.

2. If two rows (or two columns) are equal then the determinant is zero.

3. Scaling any row by λ results in a scaling of the determinant by λ

4. Interchanging rows (or columns) changes the sign of the determinant.

5. A lower or upper triangular matrix has a determinant given as product of diagonal terms.

6. The determinant of a matrix and determinant of its transpose are equal

Properties of determinants1. Adding a multiple of any row to another row does not change the

determinant.

Properties of determinants2. If two rows (or two columns) are equal then the determinant is zero.

Properties of determinants3. Scaling any row by λ results in a scaling of the determinant by λ.

Properties of determinants

4. Interchanging rows (or columns) changes the sign of the determinant.

Properties of determinants5. A lower or upper triangular matrix has a determinant given as product of diagonal terms.

Properties of determinants6. The determinant of a matrix and determinant of its transpose are equal

Find the determinant (using properties)

200

010

563

A

Check answers using MATLAB – e.g det(A)


21356104

03185

00222

0001

B

Check answers using MATLAB – e.g det(A)


241

012

561

A


21356104

73185

015123

0541

B


004

016

561

A

21356104

0005

05123

0021

B

Adjoint

The adjoint matrix is the transpose of the matrix of cofactors.

This will be used in semester 2 for solving linear simultaneous equations.

All the cofactors need to be computed! THIS IS TEDIOUS!

However: The adjoint is not needed if only the determinant is required.

matrices and matlab dr viktor fedun automatic control and systems engineering, c09 based on lectures...

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