1 / 38

MATLAB Vectors & Matrices

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

dyami
Download Presentation

MATLAB Vectors & Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATLABVectors & 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=[1 2 3]; % data assignment for row vector • or • >>p=[1,2,3] ; • >>q=[1;2;3]; %data assignment for column vector • or • >>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= 3 6 0 4 8 0 5 10 0

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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]

  10. 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)

  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=[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

  13. 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

  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 = 1 3 3 3 6 9 9 9 • Working with multidimensional arrays: Most of the concepts are similar to two dimensional arrays

  15. 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

  16. 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

  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= 0 0 • 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

  18. 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

  19. 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.

  20. 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

  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=[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

  23. 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

  24. 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

  25. 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

  26. 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

  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=[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

  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 = 0 1.0000 0.0000 • >>z=cos(x) • z = 1.0000 0.0000 -1.0000

  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=[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

  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=[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

  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',[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

  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 = 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

  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)={[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'

  35. 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

  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. =

More Related