MATLAB Vectors & Matrices

Nafees AhmedNafees AhmedAsstt. Professor, EE DepttDIT, DehraDun

Introduction

In MATLAB, matrix is chosen as a basic data element

Vector: Matrix of 1xn or nx1 is know as vector.

row vector column vector >>p=[1 2 3]; % data assignment for row vector

or >>p=[1,2,3] ;

>>q=[1;2;3]; %data assignment for column vectoror

>>q=[1 2 3];

Working with vectors

Scalar:• Matrix of 1x1 • >>r=3; %data assignment for scalar

In MATLAB it is possible to work with the complete matrix simultaneously.

• Vector Product • >>x=[3;4;5]; %column vector of 3x1• >>y=[1 2 0]; %row vector of 1x3• >>z=y*x

Z=11 %scalor

>>z=x*y %matrix of 3x3

z= 3 6 0

4 8 0

5 10 0

Working with vectors

Vector transpose

>>xt=x’ % transpose of x

>>xt= 3 4 5

>>yt=y’

yt= 1

2

0

Creating Evenly spaced row vector

>>a=1:2:11 %staring from 1 with an increment of 2 and upto 11

a =

1 3 5 7 9 11

Working with vectors

• Exercise – >>t=12.5:-2.5:0– >>t=11:1:5– >>t=1:20

• Linspace command– >>a=Linspace(0,10,5)

%linspace(x1,x2,n) , n equally spaced elements starting from x1 end with x2

– >>a=logspace(0,4,3)– %logspace(a,b,n), logarithmically spaced vector of

length n in the interval 10a to 10b

Working with vectors

• Exercise• >>x=[1 4 11 100];• >>y=[14; 200; -100];• >>z=[1.4 10.7 -1.1 20.9];• Try this • sum(x) %sum of all elements of row or column vector • mean(x) %ave of all elements of row or column vector • Max(x)• Min(x)• Prod(x) %product of all elements of row or column vector • Sign(x) %return +1 if sign of element is +ve

0 if element is zero

-1 if sign of element is -ve

• Find(x) %returns the linear indices corresponding to non-zero entries of the array x

>>a=find(x)

a= 1 2 3 4

%1st , 2nd, 3rd, & 4th, elements are non zeros>>a=find(y>24)

a= 2 %2nd element of y has value > 24

• Fix (z) %rounds the elements of a vector z to nearest integers towards zero.

• Floor(z) %rounds the elements of a vector z to nearest integers towards –infinity

• Ceil (z) %rounds the elements of a vector z to nearest intergers towards +infinity

• Round(x) %rounds the elements to nearest integer

• Sort(x, ‘mode’) %for sorting, mode=ascend or descend, default is ascend.

– >>x1=[5 -3 10 -10];– >>a=sort(x1)– >>a= -10 -3 5 10– >>b=sort(x1, ‘descend’)– >>b = 10 5 -3 10

• Mod(x,y) %Modulus after division • Rem(x,y) %Remainder after division

Working with Matrices

• Entering data in matrices – >>A=[1 10 20; 2 5 6;7 8 9]– >>B=[1+2i 3i; 4+4i 5] % i or j – >>C=[1 -2; sqrt(3) exp(1)]

• Line continuation: sometimes it is not possible to type data input on the same line– >>A=[1 10 20; 2 5 6;7 8 9] %semi column to separate rows – >>A=[1 10 20 %Enter key or carriage return

2 5 6

7 8 9]– >>A=[1, 10, 20; 2, 5,… %ellipsis(3 dots …) method

6;7, 8, 9]

Working with Matrices

• Sub-matrices– >>B=A(1:2,2:3) %row 1 to 2 & column 2 to 3 – >>B= 10 20

5 6 – >>B= A(:, 2:3) %all row & column 2 to 3 – >>B=A(:, end) %end=> last column (or row)

• Size of matrix– >>[m, n]=size(A)

Multidimensional Arrays\Matrices

• Creating multidimensional arrays: Consider a book, line no & column no represents two dimensions and third dimension is page no.

Three methods – 1. Extending matrix of lower dimension– 2. Using MATLAB function – 3. Using ‘cat’ function

Multidimensional Arrays\Matrices

• 1 Extending matrix dimension – >>A=[1 2 3; 5 4 3; 1 3 6];– >>B=[2 4 6; 1 3 6; 3 6 9];– >>A(:,:,2)=B– A(:,:,1) =

1 2 3

5 4 3

1 3 6

A(:,:,2) =

2 4 6

1 3 6

3 6 9

Multidimensional Arrays\Matrices

• 2. Using MATLAB functions– >>B=randn(4, 3, 2) %random no multidimensional matrix

B(:,:,1) = %similarly ‘ones’ & ‘zeros’ function

0.5377 0.3188 3.5784

1.8339 -1.3077 2.7694

-2.2588 -0.4336 -1.3499

0.8622 0.3426 3.0349

B(:,:,2) =

0.7254 -0.1241 0.6715

-0.0631 1.4897 -1.2075

0.7147 1.4090 0.7172

-0.2050 1.4172 1.6302

Multidimensional Arrays\Matrices

• 3. Using ‘cat’ function: concatenates a list of array – >>A1=[1 3; 6 9];– >>B1=[3 3; 9 9];– >>B=cat(2, A1, B1)

B =

1 3 3 3

6 9 9 9 Working with multidimensional arrays: Most of the concepts are similar

to two dimensional arrays

Matrix Manipulations

• Reshaping matrix into a vector – >>A=[1 10 20; 2 5 6; 7 8 9]– >>B=A(:) %converts to column matrix

B =

1

2

7

10

5

8

20

6

9

Matrix Manipulations

• Reshaping a matrix into different sized matrix

>>A=[1 2 3 4; 5 6 7 8; 9 10 11 12] % A is 3x4 matrix

>>B=reshape(A, 6,2) % reshaped matrix B is 6x2

B =

1 3

5 7

9 11

2 4

6 8

10 12

Note: total no of elements 3x4=6x2=12 must be same

Matrix Manipulations

• Expanding matrix size

>>C(2,2)=10 %D is 2x2 with last element D(2,2)=10

C=

0 0

0 10

>>D(2,1:2)=[3 4] %D is 2x2 with element D(2,1)=3 & D(2,2)=4

D=

0 0

3 4

>>A=[6 7; 8 9]; %A is 2x2 matrix

>>A(2,3)=15 %Now A is changed to 2x3 matrix

A =

6 7 0

8 9 15

Matrix Manipulations

• Appending/Deleting a row/column to a matrix

>>A=[6 7; 8 9];

>>x=[1; 2]; %Column vector

>>y=[3 4]; %Row vector

>>B=[A x] %Appending a column ‘x’

B =

6 7 1

8 9 2

>>C=[A; y] %Appending a row ‘y’

C =

6 7

8 9

3 4

Matrix Manipulations

>>C(2,:)=[ ] %delete 2nd row of matrix C

C =

6 7

3 4

>>B(:,1:2)=[ ] %delete 1st to 2nd column of matrix B

B =

1

2

Note: Deletion of single element is not allowed, we can replace it.

Matrix Manipulations

Concatenation of matrices

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

>>B=[A A+12; A+24 A+10]

B =

1 2 13 14

3 4 15 16

25 26 11 12

27 28 13 14

Generation of special Matrices Try this

>>A=zeros(2,3)

>>B=ones(3,4)

>>C=eye(3,2) %1s in main diagonal rest elements will be zero

>>D=rand(3) %3x3 matrix with random no b/w 0 to1

>>E=rands(3) %3x3 matrix with random no b/w --1 to1

>>V=vander(v) %Vandermode matrix, V whose columns are powers of the vector v. Let v=[1 2 3]. Here 3 elements => V is 3x3

Note: zeros(3,3) may be written as zeros(3) and so the others also.

Generation of special Matrices

>>d=[2 3 4 5]; %Note ‘d’ may be row/column vector

>>A=diag(d) %diagonal of A (4x4) will be 2,3,4,5 and rest ‘0’

A =2 0 0 0

0 3 0 0

0 0 4 0

0 0 0 5

>>B=diag(d,1) %1st upper diagonal elements are vector d

A =

0 2 0 0 0

0 0 3 0 0

0 0 0 4 0

0 0 0 0 5

0 0 0 0 0

>>C=diag(d,-1) %1st lower diagonal elements are vector d

Generation of special Matrices

>>x=[1 2 3; 4 5 6;7 8 9]; %Note x is a 3x3 matrix now

>>A=diag(x) %will give you the diagonal elements

A =

1

5

9

Note: diag(x,1)=>1st upper diagonal elements

diag(x,-1)=>1st lower diagonal elements

Some useful commands for matrices

• >>A=[1 2; 0 4];• >>det(A) %determinant of A• >>rank(A) %rank of A• >>trace(A) %sum of diagonal elements• >>inv(A) %inverse of A• >>norm(A) %Euclidean norm of A• >>A’ %transpose of A• >>x=A\b %left division • >>poly(A) %coefficients of characteristic equation i.e (sI-A)• >>eig(A) %gives eign values of A• >>[v,x]=eig(A) %returns v=eign vector & x=eign values • >>B=orth(A) %B will be orthogonal to A i.e. B’=B-1

• >>Find(A) %returns indices of non-zeros elements• >>sort(A) %sort each column in ascending order

Matrix and Array Operation

• Arithmetic operation on Matrix

>>A=[5 10; 15 20];

>>B=[2 4; 6 8];

Try these

>>C=A+B %or C=plus(A,B) addition

>>D=A-B %or D=minus(A,B) Subtraction

>>E=A*B %Multiplication

>>F=A^2 %Power

>>G=A/B %Right Division

>>H=A\B %Left Division

Example: Solve A.x=B where A=[2 4; 5 2] & B=[6;15]

Sol: x=A-1B=A\B

Matrix and Array Operation

• Arithmetic operation on Arrays (Element by Element Operation)

>>A=[5 10; 15 20];

>>B=[2 4; 6 8];

Try these

Note: 1. Addition and subtraction are same

2. No of Rows and Columns of two matrices must be same

>>E=A.*B % Element by Element Multiplication

>>F=A.^2 % Element by Element Power

>>G=A./B % Element by Element Right Division

>>H=A.\B % Element by Element Left Division

Rational Operators

• < Less than• <= Less than equal to • > Greater than• >= Greater than equal to • == Equal to • ~= Not equal to

• Note: true =1; false =0, try: >>6>5 on MATLAB command window

Logical Operators

• & Logical AND• | Logical OR• ~ Logical NOT, complements every element of an array• xor Logical exclusive-OR• Try these

>>x=[2 3 4];

>>y=[2 5 1];

>>x&y

>>x|y

>>m=~x % complements every element of an array

>>m=xor(x,y)

• Note: true =1; false =0, try >>6>5 on matlab command window

Function with array inputs

• If input to a function is an array then function is calculated element-by-element basis.

• Try this • >>x=[0, pi/2,pi];• >>y=sin(x)• y =

0 1.0000 0.0000• >>z=cos(x) • z =

1.0000 0.0000 -1.0000

Structure Arrays • Structure:

I. Collection of different kinds of data(text, number, numeric array etc), unlike array which contain elements of same data type.

II. Again this is 1x1 structure array • Try this

>>student.name=‘Kalpana Rawat’

>>student.rollno=44

>>student.marks=[45 33 15 18 0]

>>student

student =

name: 'Kalpana Rawat'

marks: [45 33 15 18 0]

rollno: 44

Note: Here student is structure name & name, rollno, marks are field name

Structure Arrays

Student is a 1x1 structure array having 3 fields. To increase the size of structure array define the second structure element of the array as

>>student(2).name=‘Kuldeep Rawat’;

>>student(2).rollno=57;

>>student(2).marks=[4 13 35 36 9];

>>student

student =

1x2 struct array with fields:

name

marks

rollno

Note: Here structure student will show you only field names

not filed values

Structure Arrays

• Struct function • A function struct can be used to define a structure array. Syntax is

Student=struct(‘filed1’,vaule1, ‘filed2’, value2,….)• Previous structure example can be rewritten as

>> student=struct('name','Kalpana Rawat','rollno',44,'marks',[44 34 67 19 9])

>> student(2)=struct('name',‘Mallika Rawat','rollno',45,'marks',[22 14 27 29 0])

>> student

1x2 struct array with fields:

name

rollno

marks

Note: Nesting of structure is also possible i.e filed may be another structure

Structure Arrays

• Obtaining data from structures

>>first_student_name=student(1).name

first_student_name =

Kalpana Rawat

>>first_student_rollno=student(1).rollno

first_student_rollno =

44

>>first_student_Marks=student(1).marks

first_student_Marks =

44 34 67 19 9• Try these

>>Second_student_name=student(2).name

>>Second_student_rollno=student(2).rollno

>>Second_student_Marks=student(2).marks

Cell Arrays

Cell Arrays: Array of Cells

>> sample=cell(2,2); %sample is a 2x2 cell array

Entering values in cell arrays

>> sample(1,1)={[54 37 59; 18 69 59; 72 27 49]};

>>ample(1,2)={'Mallika Tiwari'};

>>sample(2,1)={[2i,1-7i,-6]};

>>sample(2,2)={['abcd', 'efgh', 'ijkl']};

To display cell array sample in condensed form, type

>>sample

sample =

[3x3 double] 'Mallika Tiwari'

[1x3 double] 'abcdefghijkl'

Cell Arrays

To display the full cell contents use celldisp function

>> celldisp(sample)

sample{1,1} =

54 37 59

18 69 59

72 27 49

sample{2,1} =

0 + 2.0000i 1.0000 - 7.0000i -6.0000

sample{1,2} =

Mallika Tiwari

sample{2,2} =

abcdefghijkl

Cell Arrays

For graphical display use cellplot

>> cellplot(sample)

Some useful commands of structure & cell

• Cell2struct, syntax

sample_struct=cell2struct(sample, fields, dimen)

• Num2cell , syntax

c_array=num2cell(number)

• Struct2cell , syntax

c_array=struct2cell(sample-struct)

=