cz1102 computing & problem solving lecture 6

8/8/2019 CZ1102 Computing & Problem Solving Lecture 6

1/32

Matrices

Week8

ByJi Hui


2/32

Matrices

Matlab standsforMatricesLaboratory

Specifiedto

work

with

data

in

the

form

of

matrix

MeaningofMatrix

Anarrangement

of

data

in

rows

and

columns,

e.g.

atable.

Amathematicalobject,particularmathematical

operationsare

defined

in

linear

algebra:

e.g.

matrixmultiplication


3/32

Exampleof

Matrices

Matrices

Dimension:3x2. Dimension:2x4

Vectors

Specialcaseofmatrixwith

ony onerow(oronlyonecolumn

2 1

A= 0 5

4 8

0 5 3 1B =

2 0 9 6

1

2c =

3

4

d = 6 3 2


4/32

Exampleof

matrix

data

A readymix concrete company has three factories (S1, S2

and S3) which must supply three building sites (D1, D2and D3). The costs, in some suitable currency, of

transporting a load of concrete from any factory to any

site are given by the following cost table:

D1 D2 D3

S1 3 12 10

S2 17 18 15

S3 7 10 24


5/32

Reviewson

matrix

and

its

operations

Index,elementsandmatrix

,


6/32

Matrixalgebra

Transpose

,

,


7/32

(cont)

Matrixaddition

Scalemultiplication

, , ,

, ,


8/32

(cont)

Matrixmultiplication

, , ,


9/32

(cont)

Specialmatrix

Nodivisionformatrices

IfAisinvertiblesquarematrix,thereexista

matrix

,s.t.


10/32

Creatingmatrix

Creatingvector

usecomma

(or

empty

space)

to

separate

items

e.g.[1,8,4,0]or[1840]

Creatingmatrix

useasemicolon

to

indicate

the

end

of

arow

e.g.>>a=[1,2,3;4,5,6]

a=

1 2 3

4 5 6


11/32

Constructingbigmatrixfromsmallermatrices

e.g.>>

c =

[1

2;

34];

>>x=[56];

>>a=[c;x]

b=

1 2 5

3

4

6

(cont)

1 2

3

4

5 6a =

c

x=


12/32

Subscripts

Individualelementsofamatrixarereferenced

withtwo

subscripts,

the

first

for

the

row,

and

thesecondforthecolumn,

e.g.a(2,3)referselementsat2throwand3th

column>>a(2,3)

ans =

6


13/32

(cont)

Alternatively,youmayuseasingleindexby

thinkamatrix

as

being

unwounded

column

bycolumn

e.g. >>a(3)

ans

=2

1

2

3

4 5 6

1

4

25

3

6


14/32

Thecolon

operator

for

subscripts

Thecolonoperatorisextremelypowerful,and

providesfor

very

efficient

ways

of

handling

matrices

e.g.ifaisthematrix

(1) >>a(2:3,1:2)

ans =

45

78

1 2 3

4 5

67 8 9

%returnssecondandthirdrows,firstandsecond

columns


15/32

(cont)(2)>>a(2,:)

ans =7 8 9

%returnsthewholesecondrow

(3)>>a(1:2,2:3)=[0,0;0,0]

ans =1 0 0

4 0 0

7

8

9% replacesthe2by2submatrix composedofthefirstandsecondrowand thesecondandthirdcolumnwithasquare matrixof0s).

1 2 3

4 5

67 8 9


16/32

Whatishappeningusingcolon

operator

thecolonoperatorisbeingusedtocreatevector

subscripts.

However,acolonbyitselfinplaceofasubscriptdenotesalltheelementsofthecorrespondingroworcolumn.

Youcan

use

vector

subscripts

to

get

more

complicatedeffects, e.g.a(:,[13])=b(:,[42])replacesthefirstandthird

columns

of

a

by

the

fourth

and

second

columns

of

b

(a

andbmusthavethesamenumberofrows).

Whathappentoa(:)


17/32

Thekeyword

endinsubscripts.

Thekeywordend referstothelastrowor

columnof

an

array.

e.g.>>a[2,2:end]

ans =

5 6

1 2 3

4 5 6

7

8

9


18/32

Elementarymatrices

Thereisagroupoffunctionstogenerate

elementarymatrices,

See

help

elmat

thefunctionszeros(n),ones(n) andrand(n) generatenbynmatricesof1s,0sandrandomnumbers,respectively.

>>zeros(3)

ans =

0 0 0

0

0

0

0 0 0


19/32

(cont)

functioneye(n) generatesannn identitymatrix

>>eye(3)

ans =

1 0 0

0 1 0

0 0 1


20/32

Anexampletoconstructtridiagonal

matrix

>>a=2*eye(5);

>>a(1:4,

2:5)

=a(1:4,

2:5)

+eye(4);

>>a(2:5,1:4)=a(2:5,1:4)+eye(4)

a=

21000

12100

01210

00121

00012


21/32

transpose Thetransposeoperator(apostrophe)turnsrows

into

columns

and

vice

versa. e.g.>>a=[1,2,3;4,5,6]

a=

1 2 3

4

5

6>>a'

ans =

1

42 5

3 6


22/32

Changingshape

of

matrices

Thefunction reshape(x,m,n)returnsthembyn

matrix

whose

elements

are

taken

column

wise

from

x e.g.>>a=[123;456]

a=

1 2 3

4

5

6

>>reshape(a,3,2)

ans =

1

54 3

2 6

1 2

3

4 5 6

1

42

5

36

1

54 3

2 6


23/32

Manipulatingmatrices

Herearesomefunctionsformanipulatingmatrices.Seehelpfordetails.

Diag extractsorcreatesadiagonal.

Fliplr

flipsfromlefttoright.

Flipud flipsfromtoptobottom.

rot90

rotates.

tril

extractsthelowertriangularpart

triu

extractstheuppertriangularpart.


24/32

Someusefulfunctionsformatrix

information

size

Givethe

dimension

information

of

amatrix

Length

Givethenumberofelementsofavector

find

Findtheindexofnonzeroelementsofmatrix.


25/32

Multidimensionalarrays

MATLABarrayscanhavemorethantwodimensions.

e.g.>>a(:,:,1)=[1:2;3:4]

>>a(:,:,2)=[5:6;7:8]

a(:,:,1)=

1 2

3

4a(:,:,2)=

5 6

7 8

Ithelps

to

think

of

the

3D

array

aas

aseries

of

pages,

withamatrixoneachpage.Thethirddimensionofanumbersthepages


26/32

Matlab functionson

matrices

Theoperationsoneachelementsofthematrix

couldbe

different

on

the

operations

on

thematricesasamathematicalobject

Somebuiltinmatlab functionoperatesonevery

elementof

the

matrix,

as

you

would

expect

e.g.>>sin([1,2;3,4])

ans

=0.8415 0.9093

0.1411 0.7568


27/32

(cont)

However,somebuiltinmatlab functionoperates

thematrix

column

wise

e.g. >>sum([1,2;3,4])

ans=

4

6

Ifyouarenotsurewhetheraparticularfunction

operatescolumnwiseorelementbyelementon

matrices,youcanalwaysrequesthelp.

1 2

3

4


28/32

Basicmatrix

operators

Matrixaddition(elementwise)>>a=[1,2;3;4];

>>b=[5,6;0,1];

>>a+b

ans=

6 8

3

3

Matrixsubtraction(elementwise)

>>ab

ans =

4

4

3 5


29/32

(cont) Matrixmultiplication(NOTelementwise)

In

general

>>a*b

ans =

5

415 14

>>b*a

ans =23 34

3 4


30/32

Howto

do

element

wise

multiplication

Elementwisemultiplicationvs matrix

multiplication

>>a.*

b

ans =

5

120 4

, , , , , ,

1

( vs ) ( * )*n

i j i j i j i j i k k j

k

A A B a b B a b


31/32

Matrixexponential

matrixexponential(NOTelementwise)

Ashouldbeasquarematrix

>>a^3

Ifneedelementwiseexponential

>>a.^3


32/32

Someother

useful

matrix

function

det

determinant.

eig

eigenvaluedecomposition.

inv

Inversematrix

svd

singularvaluedecomposition.

cz1102 computing & problem solving lecture 6

Documents