Question

Publish using a MatLab function for the following:

If a matrix A has dimension n×n and has n linearly independent eigenvectors, it is diagonalizable.This means there exists a matrix P such that P^(−1)AP=D, where D is a diagonal matrix whose diagonal entries are made up of the eigenvalues of A. P is constructed by taking the eigenvectors of A and using them as the columns of P. Your task is to write a program (function) that does the following

If a matrix A has dimension n x n and has n linearly independent eigenvectors, it is diagonalizable. This means there exists a matrix P such that P-1AP D, where D is a diagonal matrix whose diagonal entries are made up of the eigenvalues of A. P is constructed by taking the eigenvectors of A and using them as the columns of P. Your task is to write a program (function) that does the following (a) Write a code that performs the following (you may use built-in functions): Finds the eigenvectors of an input matrix A . Checks if the eigenvectors are linearly independent. (Hint: Think determinants. You might want to use the floor function.) - if they are not linearly independent, exit the program & display error Displays P, P-1 and D (if possible) . Shows that PDP-1A that A- PDP-1. Your program should not generate the error message in part (a).) D such that B PDP-1. Your program should generate the error message in part (a).) (b) Run the program for a diagonalizable 3 x 3 matrix A. (That is, there are matrices P and D such (c) Run the program for a non-diagonalizable 3 x 3 matrix B. (That is, there are not matrices P and

Answer 1

MATLAB Script:

close all
clear
clc

fprintf('Example 1\n-----------------------------------\n')
A = [2 -1 -1; -1 2 -1; -1 -1 2];
[P,D] = diagonalize(A);

fprintf('\nExample 2\n-----------------------------------\n')
B = [0 -6 -4; 5 -11 -6; -6 9 4];
[P,D] = diagonalize(B);

function [P,D] = diagonalize(A)
disp('Input Matrix, A ='), disp(A)
[P,D] = eig(A);
if abs(real(det(P))) > eps
% Linearly independent eigenvectors => Diagonalizable
disp('P ='), disp(P)
disp('MatrixInverse(P) ='), disp(inv(P))
disp('D ='), disp(D)
disp('P*D*MatrixInverse(P) ='), disp(P*D/P)
else
error('Matrix is not diagonalizable.')
end
end

Output:

Example 1
-----------------------------------
Input Matrix, A =
2 -1 -1
-1 2 -1
-1 -1 2
P =
0.5774 0.7634 0.2895
0.5774 -0.6325 0.5164
0.5774 -0.1310 -0.8059
MatrixInverse(P) =
0.5774 0.5774 0.5774
0.7634 -0.6325 -0.1310
0.2895 0.5164 -0.8059
D =
0.0000 0 0
0 3.0000 0
0 0 3.0000
P*D*MatrixInverse(P) =
2.0000 -1.0000 -1.0000
-1.0000 2.0000 -1.0000
-1.0000 -1.0000 2.0000

Example 2
-----------------------------------
Input Matrix, A =
0 -6 -4
5 -11 -6
-6 9 4
Error using code_17>diagonalize (line 23)
Matrix is not diagonalizable.
Error in code_17 (line 11)
[P,D] = diagonalize(B);