Question

[30pts] Write a robust, efficient MATLAB script to find the eigenvalues and eigenvectors of a 2 x2 matrix input by the user. You should test out your script using the following matrices. 1::)-::) 3 2 3 1 B. 1 2 C 2 3 A- D- 4 1 2 4 4 8 -3 8 You may not use any special MATLAB tools. Instead, work symbolically and derive general expressions for the eigenvalues and the eigenvectors. For instance, you may not use the SOLVE function. If you are not sure whether or not a particular function is available to you, just ask. Include your hand calculations when you submit your solution to be graded. "Fix" each eigenvector by making the smallest component equal to 1. (Note that this is NOT the most negative.) For example, if your script tells you that one of the eigenvectors is [10;-51. the corresponding "fixed" eigenvector is [-2; 11. And if the eigenvector is (5; -101, the "fixed" version is [1;-21. Output the eigenvalues and "fixed" eigenvectors for each matrix above. You should be able to generate a second "version" of each eigenvector. Once you "fix" it as well, you can compare it to the first to make sure it is correct. (A proper check returns an array of two zeros.) You can also check your eigenvectors by making sure that Mx, x where M is the matrix you are inputting, x. is the nth eigenvector, and A is the nth eigenvalue. (Again, a proper check will return an array of zeros.) The output should be meaningful, Le, either use meaningful variable names, or use DISP commands When you have a functioning script, answer the following questions. You might need to try other matrices before you see the pattern. You can also look at the equations you have derived for insight into what is controlling the eigenvalues. (a) Under what conditions will both eigenvalues be positive? (b) Under what conditions will one eigenvalue be positive and the other be negative? (c) Under what conditions will one eigenvalue be equal to zero? (d) Under what conditions will there be only one eigenvalue? Write your answers to these questions as comments at the end of your script. Combine your hand calculation, MATLAB script, and output from the Command Window into one PDF before you submit it to Moodle for grading. NOTES: . Use DET (determinant) and TRACE (sum of the diagonal elements) to make the script more efficient. These are also very useful for answering the four questions Run your script with all four given matrices above. The hand calculation should be done symbolically, using variables for the elements of the given matrix. The hand calculation should end with expressions suitable to: (1) find the eigenvalues and then (2) find two eigenvectors for each eigenvalue (so that you can check that they are equal when "fixed"). The script should agree with your hand calculation. ece201 M-problems 19-0514.doc
can u please on matlab, i have the solution on paper.

Answer 1

EXPLANATION ::

CODE:

clc;
clear all;close all;
%% Given Data
A=[3,2;4,1];
B=[3,1;2,4];
C=[1,2 ;4,8];
D=[2,3;-3,8];
[eigenA,vectorA]=eigenandvector(A)
[eigenB,vectorB]=eigenandvector(B)
[eigenC,vectorC]=eigenandvector(C)
[eigenD,vectorD]=eigenandvector(D)
%
%% function to calculate
function [eigen,Vector]=eigenandvector(X)
I = eye(rank(X));
% Note X is a 2*2 matrix
% its eigen vales is det(X-lamda*I)=0
% that implies |(X(11)-lamda)*(X(22)-lamda)-X(12)*X(21)|=0
% which is second order equation given by
% (lamda)^2-(X(11)+X(22))*lamda+((X(11)*X(22))-(X(12)*X(21))=0
% if we compare with second order equation
a=1;
b=-(X(1,1)+X(2,2));
c=(X(1,1)*X(2,2))-(X(1,2)*X(2,1));
% whos roots are (-b(+ or -)sqrt((b*b)-(4*a*c))/(2*a)
eigen(1)=(-b+sqrt((b*b)-(4*a*c)))/(2*a);
eigen(2)=(-b-sqrt((b*b)-(4*a*c)))/(2*a);

%% Eigen Vector Calculation
% for each eigen we get one vector as
% M*V=eigen*V
% for vector 1 we get equations as
% (X-eigen*I)*V=0
% by solving

%% For First Vector
m1=(X-(eigen(1)*I));
r1(1)=-m1(1,2)/m1(1,1);
r1(2)=-m1(2,2)/m1(2,1);

if r1(1)>1
Vector1(1)=1;
Vector1(2)=r1(2)*Vector1(1);
Vector1=[Vector1(1);Vector1(2)]
else
Vector1(2)=1;
Vector1(1)=Vector1(2)/r1(2);
Vector1=[Vector1(1);Vector1(2)];
end

%% For second vector
m2=(X-(eigen(2)*I));
r2(1)=-m2(1,2)/m2(1,1);
r2(2)=-m2(2,2)/m2(2,1);

if r2(1)>1
Vector2(1)=1;
Vector2(2)=r2(2)*Vector2(1);
Vector2=[Vector2(1);Vector2(2)]
else
Vector2(2)=1;
Vector2(1)=Vector2(2)/r2(2);
Vector2=[Vector2(1);Vector2(2)];
end
  
%% LETS COMBINE BOTH VECTORS
Vector=[Vector1,Vector2];
end