1. MATLAB Vectors & Matrices
Nafees Ahmed
Asstt. Professor, EE Deptt
DIT, DehraDun

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

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=
4 8 0
5 10 0

4. Working with vectors
Vector transpose >>xt=x’ % transpose of x >>xt= >>yt=y’ yt= Creating Evenly spaced row vector >>a=1:2:11 %staring from 1 with an increment of 2 and upto 11 a =

5. Working with vectors Exercise Linspace command >>t=12.5:-2.5:0
>>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

6. Working with vectors Exercise >>x=[1 4 11 100];
>>y=[14; 200; -100];
>>z=[ ];
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
if element is zero
-1 if sign of element is -ve

7. %1st , 2nd, 3rd, & 4th, elements are non zeros
Find(x) %returns the linear indices corresponding to non-zero entries of the array x
>>a=find(x) a=
%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

8. Mod(x,y) %Modulus after division Rem(x,y) %Remainder after division
Sort(x, ‘mode’) %for sorting, mode=ascend or descend, default is ascend.
>>x1=[ ];
>>a=sort(x1)
>>a=
>>b=sort(x1, ‘descend’)
>>b =
Mod(x,y) %Modulus after division
Rem(x,y) %Remainder after division

9. Working with Matrices Entering data in matrices
>>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=[ ; ; ] %semi column to separate rows
>>A=[ %Enter key or carriage return ]
>>A=[1, 10, 20; 2, 5,… %ellipsis(3 dots …) method
6;7, 8, 9]

10. Working with Matrices Sub-matrices
>>B=A(1:2,2:3) %row 1 to 2 & column 2 to 3
>>B=
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)

11. 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

12. Multidimensional Arrays\Matrices
1 Extending matrix dimension
>>A=[ ; ; ];
>>B=[ ; ; ];
>>A(:,:,2)=B
A(:,:,1) =
A(:,:,2) =

13. Multidimensional Arrays\Matrices
2. Using MATLAB functions
>>B=randn(4, 3, 2) %random no multidimensional matrix
B(:,:,1) = %similarly ‘ones’ & ‘zeros’ function
B(:,:,2) =

14. 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 =
Working with multidimensional arrays: Most of the concepts are similar to two dimensional arrays

15. Matrix Manipulations Reshaping matrix into a vector
>>B=A(:) %converts to column matrix
B = 1 2 7 10 5 8 20 6 9

16. Matrix Manipulations Reshaping a matrix into different sized matrix
>>A=[ ; ; ] % A is 3x4 matrix
>>B=reshape(A, 6,2) % reshaped matrix B is 6x2
B =
Note: total no of elements 3x4=6x2=12 must be same

17. 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= 4
>>A=[6 7; 8 9]; %A is 2x2 matrix
>>A(2,3)=15 %Now A is changed to 2x3 matrix
A =

18. Matrix Manipulations Appending/Deleting a row/column to a matrix
>>x=[1; 2]; %Column vector
>>y=[3 4]; %Row vector
>>B=[A x] %Appending a column ‘x’
B =
>>C=[A; y] %Appending a row ‘y’
C =

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

20. Matrix Manipulations
Concatenation of matrices >>A=[1 2; 3 4]; >>B=[A A+12; A+24 A+10] B =

21. 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.

22. Generation of special Matrices
>>d=[ ]; %Note ‘d’ may be row/column vector
>>A=diag(d) %diagonal of A (4x4) will be 2,3,4,5 and rest ‘0’
A =
>>B=diag(d,1) %1st upper diagonal elements are vector d
A =
>>C=diag(d,-1) %1st lower diagonal elements are vector d

23. Generation of special Matrices
>>x=[ ; ; ]; %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

24. Some useful commands for matrices
>>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

25. Matrix and Array Operation
Arithmetic operation on Matrix
>>A=[5 10; ];
>>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

26. Matrix and Array Operation
Arithmetic operation on Arrays (Element by Element Operation)
>>A=[5 10; ];
>>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

27. 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

28. Logical Operators & Logical AND | Logical OR
~ Logical NOT, complements every element of an array
xor Logical exclusive-OR
Try these
>>x=[ ];
>>y=[ ];
>>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

29. 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 =
>>z=cos(x)
z =

30. Structure Arrays Structure:
Collection of different kinds of data(text, number, numeric array etc), unlike array which contain elements of same data type.
Again this is 1x1 structure array
Try this
>>student.name=‘Kalpana Rawat’
>>student.rollno=44
>>student.marks=[ ]
>>student
student =
name: 'Kalpana Rawat'
marks: [ ]
rollno: 44
Note: Here student is structure name & name, rollno, marks are field name

31. 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=[ ]; >>student student = 1x2 struct array with fields: name marks rollno Note: Here structure student will show you only field names not filed values

32. 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',[ ])
>> student(2)=struct('name',‘Mallika Rawat','rollno',45,'marks',[ ])
>> student
1x2 struct array with fields:
name
rollno
marks
Note: Nesting of structure is also possible i.e filed may be another structure

33. 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 =
Try these
>>Second_student_name=student(2).name
>>Second_student_rollno=student(2).rollno
>>Second_student_Marks=student(2).marks

34. 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)={[ ; ; ]}; >>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'

35. Cell Arrays To display the full cell contents use celldisp function
>> celldisp(sample)
sample{1,1} =
sample{2,1} =
i i
sample{1,2} =
Mallika Tiwari
sample{2,2} =
abcdefghijkl

36. Cell Arrays
For graphical display use cellplot >> cellplot(sample)

37. 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)

38. =
Hello