Convertion of Quadratic form to Cononical form

Using MATLAB code to transform the quadratic form 3*(x1)^2 + 5*(x2)^2 + 3*(x3)^2 − 2*(x2)*(x3) +2*(x3)*(x1) − 2*(x1)*(x2) to canonical form and specify the matrix of transformation.

[Debasish Samal]

Hello Sai,

You can use the Symbolic math toolbox to solve this problem.

>> Q = 3*(x1)^2 + 5*(x2)^2 + 3*(x3)^2 - 2*(x2)*(x3) +2*(x3)*(x1) - 2*(x1)*(x2);

>> X = [x1 x2 x3];

>> H = hessian(Q)/2;

H =

[ 3, -1, 1]

[ -1, 5, -1]

[ 1, -1, 3]

Please refer the URL below for proper explantion:

https://www.mathworks.com/matlabcentral/answers/445266-polynomial-to-matrix-form-canonical-form?s_tid=answers_rc1-2_p2_MLT

Answer 1

[thode Saiprajwal]

clc
clear
syms x1 x2 x3 y1 y2 y3
q=input('enter a quadratic form in terms of x1,x2,x3');
a11=(diff(diff(q,x1),x1))/2;
a22=(diff(diff(q,x2),x2))/2;
a33=(diff(diff(q,x3),x3))/2;
a12=(diff(diff(q,x1),x2))/2;
a13=(diff(diff(q,x1),x3))/2;
a23=(diff(diff(q,x2),x3))/2;
A=[a11,a12,a13;a12,a22,a23;a13,a23,a33];
[m,d]=eig(A);
disp('eigen values of A are :')
disp(d)
disp('orthogonal matrix:')
disp(m)
disp('canonical form of q is :')
disp(d(1,1)*(y1)^2+d(2,2)*(y2)^2+d(3,3)*(y3)^2)

[Braham Aggarwal]

Thanks man

Answer 2

[Jayashree]

clc 
clear 
syms x1 x2 x3 y1 y2 y3 
%q=input('enter a quadratic form in terms of x1,x2,x3'); 
q=3*(x1)^2 + 5*(x2)^2 + 3*(x3)^2 - 2*(x2)*(x3)+2*(x3)*(x1) - 2*(x1)*(x2);
a11=(diff(diff(q,x1),x1))/2; 
a22=(diff(diff(q,x2),x2))/2; 
a33=(diff(diff(q,x3),x3))/2; 
a12=(diff(diff(q,x1),x2))/2; 
a13=(diff(diff(q,x1),x3))/2; 
a23=(diff(diff(q,x2),x3))/2; 
A=[a11,a12,a13;a12,a22,a23;a13,a23,a33]; 
[m,d]=eig(A); 
disp('eigen values of A are :') 

eigen values of A are :

disp(d) 

disp('orthogonal matrix:') 

orthogonal matrix:

disp(m) 

disp('canonical form of q is :') 

canonical form of q is :

disp(d(1,1)*(y1)^2+d(2,2)*(y2)^2+d(3,3)*(y3)^2)