MATLAB Vectors & Matrices 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]; or >>q=[1 2 3]; %data assignment for column vector

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]; >>y=[1 2 0]; >>z=y*x Z=11

>>z=x*y z= 3 %column vector of 3x1 %row vector of 1x3 %scalor %matrix of 3x3 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 of the array x >>a=find(x) a= 1 2 3

entries 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) Rem(x,y) %Modulus after division %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) >>B= 1020

5 6 >>B= A(:, 2:3) >>B=A(:, end) Size of matrix >>[m, n]=size(A) %row 1 to 2 & column 2 to 3 %all row & column 2 to 3 %end=> last column (or row) 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 >>D(2,2)=10 %D is 2x2 with last element D(2,2)=10 D= 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,:)=[ ] C= 6 7 3 4 >>B(:,1:2)=[ ] B= 1 2 %delete 2nd row of matrix C %delete 1st to 2nd column of matrix B

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

< <= > >= == ~= Note: true =1; false =0, try: >>6>5 on MATLAB command window Less than Less than equal to Greater than Greater than equal to

Equal to Not equal to 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-byelement 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]}; >>sample(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) =