Polynomial to Matrix form(canonical form)

How to convert the given quadratic form(Q = x1^2 + 2x1x2+x2^2) into its canonical form in matlab.

[Walter Roberson]

factor() ?

[Sudhish Subramaniam]

Sir can you please elaborate on the answer ?

I mean can you please send the code .

[madhan ravi]

write your expected result explicitly

[Sai Teja Sunkari]

what about 3x3 matrix

[Walter Roberson]

https://www.mathworks.com/matlabcentral/answers/445266-polynomial-to-matrix-form-canonical-form#answer_470380

Answer 1

[lalith keerthan]

syms x1 x2 x3 y1 y2 y3 a b c p
Q=input('Enter the form in x1 x2 x3')
a11=(1/2)*diff(diff(Q,x1),x1)
a22=(1/2)*diff(diff(Q,x2),x2)
a33=(1/2)*diff(diff(Q,x3),x3)
a12=(1/2)*diff(diff(Q,x1),x2)
a21=a12
a13=(1/2)*diff(diff(Q,x1),x3)
a13=a31
a23=(1/2)*diff(diff(Q,x2),x3)
a23=a23
A=[a11,a12,a13;a21,a22,a23;a31,a32,a33]
[N D]=eig(A)
X=[x1,x2,x3]
Y=[y1,y2,y3]
disp(D(1,1)*y1^2+D(2,2)*y2^2+D(3,3)*y3^2)
[m,n]=size(A);
for i=1:n
N(:,i)=[N(1,i)/sqrt(N(1,i)^2+N(2,i)^2+N(3,i)^2) N(2,i)/sqrt(N(1,i)^2+N(2,i)^2+N(3,i)^2) N(3,i)/sqrt(N(1,i)^2+N(2,i)^2+N(3,i)^2)]
end
display('no repeated eigen value and the orthogonal transformation is X=NY')
X==(N*Y)

[Walter Roberson]

Is that a question or a solution to the poster's question ?

[John D'Errico]

I think it was an attempt at an answer/ At least it started out as one, sort of. But things got lost along the way, following a convoluted, confused path at the end.

Answer 2

[John D'Errico]

I assume the question is to resolve a quadratic polynomial, perhaps:

Q = x1^2 + 2*x1*x2 + x2^2

into a quadratic form. That is, given Q, you want to recover the matrix H, such that

Q = [x1,x2]*H*[x1;x2]

This is quite easy using the symbolic toolbox. The desired matrix H is 1/2 times the Hessian matrix of Q.

For example, given the quadratic Q...

syms x1 x2
Q = x1^2 + 2*x1*x2 + x2^2
Q =
x1^2 + 2*x1*x2 + x2^2

X = [x1,x2];
H = hessian(Q)/2
H =
[ 1, 1]
[ 1, 1]

H is the desired matrix. We can see Q is recovered:

expand(X*H*X.')
ans =
x1^2 + 2*x1*x2 + x2^2

This is just an educated guess on my part as to the answer. Since there has been no response from the OP since it as first posted, we can only guess.

[Gaurav Malik]

Yeah but what do you do when you also have linear terms in your function? We need also the c'X term.

[Walter Roberson]

@Gaurav Malik

In such a case are you working with a quadratic form ? Are you, as John indicates, trying to recover the H in Q = [x1,x2]*H*[x1;x2] ?

Answer 3

[Walter Roberson]

syms x1 x2

QQ = x1^2 +2*x1*x2+x2^2

factor(QQ)