Lesson 4

Arrays in MATLAB First round Lecturer : Dr. Noam Amir

Format Revised. 01-11-2007

CE2002-NTUST

Outline:

Testing arrays in MATLAB The colon operator Concatenating arrays in MATLAB Array indexing Array math Sorting and searching in arrays

Working with Matrices

MATLAB == MATrix LABoratory

>> load durer >> image(X) >> colormap(map)

>> load detail >> image(X) >> colormap(map)

Every variable is an array:

>> a=1;

>> size(a) ans = 1 1

>> length(a) ans = 1

>> a=1;

>> size(a) ans = 1 1

>> length(a) ans = 1

Even a scalar

Checking its dimensions:

Checking on the number of elements:

>> a=[1 2 3]

a=

1 2 3

>> size(a) ans = 1 3

>> length(a) ans = 3

>> a=[1 2 3]

a=

1 2 3

>> size(a) ans = 1 3

>> length(a) ans = 3

A vector can be a row or a column:

>> a=[1; 2; 3]

a=

1

2

3

>> size(a) ans = 3 1

>> length(a) ans = 3

>> a=[1; 2; 3]

a=

1

2

3

>> size(a) ans = 3 1

>> length(a) ans = 3 row

column

Entering Numeric Arrays

>> a=[1 2;3 4]

a =

1 2

3 4

>> b = 2:-0.5:0 b = 2 1.5 1 0.5 0

>> c = rand(2,4) c = 0.8913 0.45647 0.82141 0.61543 0.7621 0.018504 0.4447 0.79194

>> a=[1 2;3 4]

a =

1 2

3 4

>> b = 2:-0.5:0 b = 2 1.5 1 0.5 0

>> c = rand(2,4) c = 0.8913 0.45647 0.82141 0.61543 0.7621 0.018504 0.4447 0.79194

Row separator: semicolon (;) Column separator:space / comma (,)

Use square brackets [ ]

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

Creating sequences using the colon operator (:)

Utility function for creating matrices. (Ref: Utility Commands)

Entering Numeric Arrays - cont.

>> w = [-2.8, sqrt(-7), (3+5+6)*3/4] w = -2.8 0 + 2.6458i 10.5 >> x(3,2) = 3.5 x = 0 0 0 0 0 3.5

>> w(2,5) = 23

w = -2.8 0 + 2.6458i 10.5 0 0 0 0 0 0 23

>> w = [-2.8, sqrt(-7), (3+5+6)*3/4] w = -2.8 0 + 2.6458i 10.5 >> x(3,2) = 3.5 x = 0 0 0 0 0 3.5

>> w(2,5) = 23

w = -2.8 0 + 2.6458i 10.5 0 0 0 0 0 0 23

Using other MATLAB expressions

Matrix element assignment

Any MATLAB expression can be entered as a matrix element

Some more about the colon (:) :

>> 1:5 ans = 1 2 3 4 5 >> a=5:1 a = Empty matrix: 1-by-0

>> 1:0.3:2

ans = 1.0000 1.3000 1.6000 1.9000

>> 1:5 ans = 1 2 3 4 5 >> a=5:1 a = Empty matrix: 1-by-0

>> 1:0.3:2

ans = 1.0000 1.3000 1.6000 1.9000

Default step is 1

Impossible values are not an error

If upper limit is not attainable – the values stop before it

Alternatives to the colon

>> linspace(0,pi,5) ans= 0 0.7854 1.5708 2.3562 3.1416 >> logspace(0,2,3) ans= 1 10 100 >> logspace(0,1,10) ans= Columns 1 through 6 1.0000 1.2915 1.6681 2.1544 2.7826 3.5938 Columns 7 through 10 4.6416 5.9948 7.7426 10.0000

>> linspace(0,pi,5) ans= 0 0.7854 1.5708 2.3562 3.1416 >> logspace(0,2,3) ans= 1 10 100 >> logspace(0,1,10) ans= Columns 1 through 6 1.0000 1.2915 1.6681 2.1544 2.7826 3.5938 Columns 7 through 10 4.6416 5.9948 7.7426 10.0000

• linspace and logspace:

• linspace(first,last,N)

• logspace(first_exp,last_exp,N)

Numerical Array Concatenation - [ ]

>> a=[1 2;3 4]

a =

1 2

3 4

>> cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24

>> a=[1 2;3 4]

a =

1 2

3 4

>> cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24

Use [ ] to combine existing arrays as matrix “elements”

Use square brackets [ ]

4*a

Row separator: semicolon (;) Column separator:space / comma (,)

Matrices must be rectangular.

Indexing into a vector:

>> a=5:10 a = 5 6 7 8 9 10 >> a(1) ans= 5 >> a(2:4) ans= 6 7 8 >> a([1 3 2]) ans= 5 7 6

>> a=5:10 a = 5 6 7 8 9 10 >> a(1) ans= 5 >> a(2:4) ans= 6 7 8 >> a([1 3 2]) ans= 5 7 6

• Indexing is done with parentheses

• A vector of indices is permissible

• Indexing is similar for row or column vectors

>> a=(5:10)’ a = 5 6 7 8 9 10 >> a(1) ans= 5 >> a(2:4) ans= 6 7 8 >> a([1 3 2]) ans= 5 7 6

>> a=(5:10)’ a = 5 6 7 8 9 10 >> a(1) ans= 5 >> a(2:4) ans= 6 7 8 >> a([1 3 2]) ans= 5 7 6

Transpose

A matrix has more than one dimension - So an index into a matrix has to address each dimension

separately, e.g.: mat(3,5)rowcolumn

BUT: After all, a matrix is stored in memory as a string of

elements And we can address that string of elements as a

VECTOR! In other words – matrices in Matlab have schizophrenia…

let’s have a look:

Indexing into a matrix:

Indexing the Matrix in MATLAB

4 10 1 6 2

8 1.2 9 4 25

7.2 5 7 1 11

0 0.5 4 5 56

23 83 13 0 10

1 2 Rows (m) 3 4

5

Columns (n)

1 2 3 4 51 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25

A =A (2,4)

A (17)

Rectangular Matrix: Scalar: 1-by-1 array Vector: m-by-1 array

1-by-n array Matrix: m-by-n array

Matrix elements can be EITHER numbers OR characters

Array Subscripting / Indexing

4 10 1 6 2

8 1.2 9 4 25

7.2 5 7 1 11

0 0.5 4 5 56

23 83 13 0 10

1 2 3 4 5

1 2 3 4 51 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25

A =

A(3,1) A(3)

A(1:5,5) A(:,5) A(21:25)

A(4:5,2:3) A([9 14;10 15])

• Use () parentheses to specify index • colon operator (:) specifies range / ALL • [ ] to create matrix of index subscripts • 'end' specifies maximum index value

A(1:end,end) A(:,end) A(21:end)’

Matrix examples:

>> a=[1 2;3 4] a = 1 2 3 4 >> a(1,4)=5 % auto enlarging a = 1 2 0 5 3 4 0 0 >> a(:,3)=[4; 4] a = 1 2 4 5 3 4 4 0 >> b=a(:,4:-1:1) b = 5 4 2 1 0 4 4 3

>> a=[1 2;3 4] a = 1 2 3 4 >> a(1,4)=5 % auto enlarging a = 1 2 0 5 3 4 0 0 >> a(:,3)=[4; 4] a = 1 2 4 5 3 4 4 0 >> b=a(:,4:-1:1) b = 5 4 2 1 0 4 4 3

>> b(6) % columnwise unraveling

ans =

4

>> b(:) % columnwise unraveling

ans =

5

0

4

4

2

4

1

3

>> b(1,:)=[] % erasing a row

b =

0 4 4 3

>> b(6) % columnwise unraveling

ans =

4

>> b(:) % columnwise unraveling

ans =

5

0

4

4

2

4

1

3

>> b(1,:)=[] % erasing a row

b =

0 4 4 3

Matrices as vectors and vice versa:

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

a =

1 2 3

4 5 6

>> a(1:2)

ans =

1 4

>> a(5)

ans =

3

>> a(end)

ans =

6

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

a =

1 2 3

4 5 6

>> a(1:2)

ans =

1 4

>> a(5)

ans =

3

>> a(end)

ans =

6

• The previous pages showed that we can index into a matrix like a vector

• Unraveling is columnwise

>> sub2ind(size(a),2,3)

ans =

6

>> [r,c]=ind2sub(size(a),4)

r =

2

c =

2

>> sub2ind(size(a),2,3)

ans =

6

>> [r,c]=ind2sub(size(a),4)

r =

2

c =

2

• 2 functions translate between the two types of indices:

Logical indexing:

>> a=-3:3 a = -3 -2 -1 0 1 2 3 >> abs(a)>1 ans = 1 1 0 0 0 1 1 >> a(abs(a)>1) ans = -3 -2 2 3

>> a=-3:3 a = -3 -2 -1 0 1 2 3 >> abs(a)>1 ans = 1 1 0 0 0 1 1 >> a(abs(a)>1) ans = -3 -2 2 3

• Logical operations give 0 or 1

• The result of such operations is alogical array

• This kind of array can be used similarly to indexes - to retain or eliminate elements of a vector:

Logical indexing – cont.:

>> a([1 1 0 0 0 1 1]) % ouch!

??? Subscript indices must either be real positive integers or logicals.

>> a(logical([1 1 0 0 0 1 1]))

ans =

-3 -2 2 3

>> a([1 1 0 0 0 1 1]) % ouch!

??? Subscript indices must either be real positive integers or logicals.

>> a(logical([1 1 0 0 0 1 1]))

ans =

-3 -2 2 3

• An array that is composed of 1’s and 0’s isn’t necessarily a logical array

• A numeric array of 1 and 0 can be converted into a logical array using: logical()

Standard matrices:

• Some useful matrices can be created more easily:

>> ones(2) ans = 1 1 1 1 >> ones(2,3) ans= 1 1 1 1 1 1 >> rand(1,4) ans= 0.4565 0.0185 0.8214 0.4447 >> diag([ 1 2 3]) ans = 1 0 0 0 2 0 0 0 3

>> ones(2) ans = 1 1 1 1 >> ones(2,3) ans= 1 1 1 1 1 1 >> rand(1,4) ans= 0.4565 0.0185 0.8214 0.4447 >> diag([ 1 2 3]) ans = 1 0 0 0 2 0 0 0 3

Makes a diagonal matrix from a vector

diag

Random numbers, gaussian distribution, zero mean, unit var.

randn

Random numbers, uniform distribution on [0,1]

rand

Identity matrixeye

All 0zeros

All 1ones

Size and length again:

>> a=ones(3,4); >> size(a) ans = 3 4 >> size(a,1) ans = 3 >> size(a,2) ans = 4

>> a=ones(3,4); >> size(a) ans = 3 4 >> size(a,1) ans = 3 >> size(a,2) ans = 4

• size gives a vector

• But we can obtain one element from it:

>> a=ones(1,5); % row

>> length(a)

ans =

5

>> a=ones(5,1); % column

>> length(a)

ans =

5

>> a=ones(5,3,4); % 3D

>> length(a)

ans =

5

>> a=ones(1,5); % row

>> length(a)

ans =

5

>> a=ones(5,1); % column

>> length(a)

ans =

5

>> a=ones(5,3,4); % 3D

>> length(a)

ans =

5

• length gives the length of a vector…

• But the maximum dimension of a matrix

Empty arrays:

>> a=ones(3,4); >> numel(a) ans = 12 >> a=[] % empty variable a = [] >> numel(a) ans = 0 >> size(a) ans = 0 0

>> a=ones(3,4); >> numel(a) ans = 12 >> a=[] % empty variable a = [] >> numel(a) ans = 0 >> size(a) ans = 0 0

• numel gives the number of elements in an array:

>> a=zeros(0,3)

a =

Empty matrix: 0-by-3

>> size(a)

ans =

0 3

>> length(a)

ans =

0

>> numel(a)

ans =

0

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

a =

1 2 3

>> a=zeros(0,3)

a =

Empty matrix: 0-by-3

>> size(a)

ans =

0 3

>> length(a)

ans =

0

>> numel(a)

ans =

0

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

a =

1 2 3

• BUT- an empty matrix can have a nonzero dimension:

Rearranging the dimensions:

>> a=1:10; % 10 elements >> reshape(a,2,5) % 10 elements! ans = 1 3 5 7 9 2 4 6 8 10 >> reshape(a,5,2) % 10 elements! ans = 1 6 2 7 3 8 4 9 5 10

>> a=1:10; % 10 elements >> reshape(a,2,5) % 10 elements! ans = 1 3 5 7 9 2 4 6 8 10 >> reshape(a,5,2) % 10 elements! ans = 1 6 2 7 3 8 4 9 5 10

• reshape takes the elements of an array and rearranges with new dimensions:

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

>> repmat(a,1,2)

ans =

1 2 1 2

3 4 3 4

>> repmat(a,[1 2]) % same

ans =

1 2 1 2

3 4 3 4

>> repmat(a,2) % ‘2’ is extended

ans =

1 2 1 2

3 4 3 4

1 2 1 2

3 4 3 4

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

>> repmat(a,1,2)

ans =

1 2 1 2

3 4 3 4

>> repmat(a,[1 2]) % same

ans =

1 2 1 2

3 4 3 4

>> repmat(a,2) % ‘2’ is extended

ans =

1 2 1 2

3 4 3 4

1 2 1 2

3 4 3 4

• repmat repeats an array

A word about transposing:

The operator for transposing is the apostrophe (‘) Complex numbers

are conjugated!

For non-conjugating transpose, use: (.’)

>> a=(5:7)+j a = 5.0000 + 1.0000i 6.0000 + 1.0000i

7.0000 + 1.0000i >> a’ ans= 5.0000 - 1.0000i 6.0000 - 1.0000i 7.0000 - 1.0000i >> a.’ ans= 5.0000 + 1.0000i 6.0000 + 1.0000i 7.0000 + 1.0000i

>> a=(5:7)+j a = 5.0000 + 1.0000i 6.0000 + 1.0000i

7.0000 + 1.0000i >> a’ ans= 5.0000 - 1.0000i 6.0000 - 1.0000i 7.0000 - 1.0000i >> a.’ ans= 5.0000 + 1.0000i 6.0000 + 1.0000i 7.0000 + 1.0000i

Array sorting Matrices: each column is sorted separately:

>> x=randperm(6) x = 2 4 3 6

5 1 >> xs=sort(x) %ascending xs = 1 2 3 4

5 6 >> [xs,ixs]=sort(x)

%ascending xs = 1 2 3 4

5 6 ixs = 6 1 3 2

5 4

>> x=randperm(6) x = 2 4 3 6

5 1 >> xs=sort(x) %ascending xs = 1 2 3 4

5 6 >> [xs,ixs]=sort(x)

%ascending xs = 1 2 3 4

5 6 ixs = 6 1 3 2

5 4

• Vectors:

>> m=[randperm(4);randperm(4);randperm(4)]

m =

1 4 3 2

2 3 4 1

1 2 4 3

>> [sm,ism]=sort(m)

sm =

1 2 3 1

1 3 4 2

2 4 4 3

ism =

1 3 1 2

3 2 2 1

2 1 3 3

>> m=[randperm(4);randperm(4);randperm(4)]

m =

1 4 3 2

2 3 4 1

1 2 4 3

>> [sm,ism]=sort(m)

sm =

1 2 3 1

1 3 4 2

2 4 4 3

ism =

1 3 1 2

3 2 2 1

2 1 3 3

Note: randperm(n) gives a random permutationof the integers from 1 to n

Array sorting – contd.

>> m

m =

1 4 3 2

2 3 4 1

1 2 4 3

>> [tmp,ind]=sort(m(:,2));

>> % based on 2'nd column

>> m(ind,:)

ans =

1 2 4 3

2 3 4 1

1 4 3 2

>> m

m =

1 4 3 2

2 3 4 1

1 2 4 3

>> [tmp,ind]=sort(m(:,2));

>> % based on 2'nd column

>> m(ind,:)

ans =

1 2 4 3

2 3 4 1

1 4 3 2

• Sorting all columns based on only one:

>> [sm,ism]=sort(m,2) % add a dimension

sm =

1 2 3 4

1 2 3 4

1 2 3 4

ism =

1 4 3 2

4 1 2 3

1 2 4 3

>> [sm,ism]=sort(m,2) % add a dimension

sm =

1 2 3 4

1 2 3 4

1 2 3 4

ism =

1 4 3 2

4 1 2 3

1 2 4 3

• Sort rows:

Searching in a vector:

>> x=-3:3

x =

-3 -2 -1 0 1 2 3

>> k=find(abs(x)>1) % find indices where abs(x)>1

k =

1 2 6 7

>> x(k) % extract those values

ans =

-3 -2 2 3

>> x(abs(x)>1) % logical addressing does the same thing

ans =

-3 -2 2 3

>> x=-3:3

x =

-3 -2 -1 0 1 2 3

>> k=find(abs(x)>1) % find indices where abs(x)>1

k =

1 2 6 7

>> x(k) % extract those values

ans =

-3 -2 2 3

>> x(abs(x)>1) % logical addressing does the same thing

ans =

-3 -2 2 3

• Searching in a vector is simple - find

Searching in a matrix:

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

m =

1 2 3 4

5 6 7 8

>> [ii,jj]=find(m>3)%one way

ii =

2

2

2

1

2

jj =

1

2

3

4

4

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

m =

1 2 3 4

5 6 7 8

>> [ii,jj]=find(m>3)%one way

ii =

2

2

2

1

2

jj =

1

2

3

4

4

• Same command - find >> ind=find(m>3)% another way

ind =

2

4

6

7

8

>> m(ind)=0 %replace with zeros

m =

1 2 3 0

0 0 0 0

>> m(ii,jj) % ouch!!

ans =

5 6 7 8 8

5 6 7 8 8

5 6 7 8 8

1 2 3 4 4

5 6 7 8 8

>> ind=find(m>3)% another way

ind =

2

4

6

7

8

>> m(ind)=0 %replace with zeros

m =

1 2 3 0

0 0 0 0

>> m(ii,jj) % ouch!!

ans =

5 6 7 8 8

5 6 7 8 8

5 6 7 8 8

1 2 3 4 4

5 6 7 8 8

maximum and minimum in a vector:>> v=rand(1,6)

v =

0.6038 0.2722 0.1988 0.0153 0.7468 0.4451

>> max(v) % maximum

ans =

0.7468

>> [mx,ind]=max(v) % maximum and index

mx =

0.7468

ind =

5

>> min(v) % minimum

ans =

0.0153

>> [mn,ind]=min(v) % minimum and index

mn =

0.0153

ind =

4

>> v=rand(1,6)

v =

0.6038 0.2722 0.1988 0.0153 0.7468 0.4451

>> max(v) % maximum

ans =

0.7468

>> [mx,ind]=max(v) % maximum and index

mx =

0.7468

ind =

5

>> min(v) % minimum

ans =

0.0153

>> [mn,ind]=min(v) % minimum and index

mn =

0.0153

ind =

4

maximum and minimum in a matrix:>> m=rand(3,6) % maximum per column

m =

0.9318 0.8462 0.6721 0.6813 0.5028 0.3046

0.4660 0.5252 0.8381 0.3795 0.7095 0.1897

0.4186 0.2026 0.0196 0.8318 0.4289 0.1934

>> [mx,r]=max(m)

mx =

0.9318 0.8462 0.8381 0.8318 0.7095 0.3046

r =

1 1 2 3 2 1

>> max(max(m)) % overall maximum – bad way

ans =

0.9318

>> [mmx,ii]=max(m(:)) % overall maximum – good way, with index!

mmx =

0.9318

ii =

1

>> m=rand(3,6) % maximum per column

m =

0.9318 0.8462 0.6721 0.6813 0.5028 0.3046

0.4660 0.5252 0.8381 0.3795 0.7095 0.1897

0.4186 0.2026 0.0196 0.8318 0.4289 0.1934

>> [mx,r]=max(m)

mx =

0.9318 0.8462 0.8381 0.8318 0.7095 0.3046

r =

1 1 2 3 2 1

>> max(max(m)) % overall maximum – bad way

ans =

0.9318

>> [mmx,ii]=max(m(:)) % overall maximum – good way, with index!

mmx =

0.9318

ii =

1

Further manipulations:

If any dimension is 1 – dimension is removedsqueeze

Extract lower diagonal from matrix (zero all that is above diagonal)

tril

Extract upper diagonal from matrix (zero all that is below diagonal)

triu

Extract diagonal from matrixdiag(m)

Make vector into a diagonal matrixdiag(vec)

Rotate 90 degrees twice counterclockwiserot(m,2)

Rotate 90 degrees counterclockwiserot90

Flip array in left-right directionfliplr

Flip array in up-down directionflipud

Array math in Matlab Linear algebra vs. simple algebra

When performing mathematical operations between arrays in Matlab – how do we determine what rules will be used? Matlab has a set of rules that is generally simple and easy to follow Let’s examine the possible cases:

Scalar/Matrix math:

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

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

Scalar expansion

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

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

Generally this works for + - * / ^ …

Similarly:

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

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

>> w=[1 2;3 4] / 5 w = 0.2 0.4 0.6 0.8

>> w=[1 2;3 4] / 5 w = 0.2 0.4 0.6 0.8

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

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

BUT:>> w=5 / [1 2;3 4] ??? Error using ==> mrdivide Matrix dimensions must agree.

>> w=5 / [1 2;3 4] ??? Error using ==> mrdivide Matrix dimensions must agree.

To resolve this we have to go a bit further…

Simple Matrix/Matrix math:

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

6 7 = 8 9

>> w=[1 2;3 4] .* [5 5;5 5]

5 10 = 15 20

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

6 7 = 8 9

>> w=[1 2;3 4] .* [5 5;5 5]

5 10 = 15 20

• Performing calculations element by element:

• Matrices must be the same size

• With + and – there is no ambiguity

• With other operators – add a dot: .* ./ .^ etc.

>> w=5 ./ [1 2;3 4] 5 2.5

1.6667 1.25

>> w=5 ./ [1 2;3 4] 5 2.5

1.6667 1.25

… and to address the problem on the previous slide:

In this case:

Matrices must have the same dimensions imensions of resulting matrix = dimensions of

multiplied matrices Resulting elements = product of corresponding

elements from the original matrices Same rules apply for other array operations

>> 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)

NEXT:Matrix Multiplication and Division

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 7 8];

>> b = ones(4,3);

>> c = a*b

c =

10 10 10 26 26 26

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

>> b = ones(4,3);

>> c = a*b

c =

10 10 10 26 26 26

[2x4]

[4x3]

[2x4]*[4x3] [2x3]

a(2nd row).b(3rd column)

Division (solving equations):using “Left Division”

To Solve the set of simultaneous equations:

we need to calculate:

>> 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

bAx 1 x b A

Some Matrix and Array Operators:

>> help ops >> help matfun

(In order of precedence)

Matrix Operators Array operators

() parentheses

’ comp. transpose .’ array transpose

^ power .^ array power

* multiplication .* array mult.

/ division ./ array division

\ left division

+ addition

- subtraction

Common Matrix Functions inv matrix inverse

det determinant

rank matrix rank

eig eigenvectors & values

svd singular value dec.

norm matrix / vector norm

• In most languages – you have to use loops:

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• In MATLAB - use Array Operations instead:

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• This kind of code is shorter and simpler

IMPORTANT Example: Array Operations

>> tic; for I = 1:10000 Density(I) = Mass(I)/(Length(I)*Width(I)*Height(I)); end; toc elapsed_time = 4.7260

>> tic; for I = 1:10000 Density(I) = Mass(I)/(Length(I)*Width(I)*Height(I)); end; toc elapsed_time = 4.7260

>> tic; Density = Mass./(Length.*Width.*Height); toc

elapsed_time =

0

>> tic; Density = Mass./(Length.*Width.*Height); toc

elapsed_time =

0

Use TIC and TOC to measure elapsed time

Vectorized code WAS much faster than loops

Before JIT!

Another example of loopless vs loopy code:

Successive differences: Cumulative sum:

>> a=1:10;

>> for iii=1:9

b(iii)=a(iii+1)-a(iii);

end >> b b = 1 1 1 1 1 1 1 1 1

>> a=1:10;

>> for iii=1:9

b(iii)=a(iii+1)-a(iii);

end >> b b = 1 1 1 1 1 1 1 1 1

>> diff(a) ans = 1 1 1 1 1 1 1 1 1

>> diff(a) ans = 1 1 1 1 1 1 1 1 1

>> cumsum(b) ans = 1 2 3 4 5 6 7 8 9

>> cumsum(b) ans = 1 2 3 4 5 6 7 8 9

loopedloopless

More about programming with arrays in a later lesson!

Summary:

Arrays are very easy to use in MATLAB No declarations are necessary, and array dimensions can be modified dynamically Manipulating and indexing into arrays is very powerful, and can be complicated at times Arrays can be sorted and searched easily Math with arrays can be performed either element by element, or as defined by the rules of linear algebra