Why my symbolic result of eigenvalue doesn't match my numeric result?

Question

JingChong Ning 2023-1-31

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1904050-why-my-symbolic-result-of-eigenvalue-doesn-t-match-my-numeric-result

回答： Sulaymon Eshkabilov 2023-1-31

在 MATLAB Online 中打开

I have a set of three points,

. As well as a matrix

[2, 0, 0]

[0, 2, 4*z2]

[0, 4*z2, 4*y2 + 2]

I need to find the eigenvalue of this matrix both symbolically and numerically after we substitude the y and z value in the matrix with the values in the provided points. However, if I plug in the numeric y and z value into the symbolic result I got from using

eigen = eig(eqn2ma)

It is different from first substituting the y and z into the matrix, then taking its eigenvalue.

This is the code

syms x y x1 y1 a x2 y2 z2 lamb
%{
%Q2
t=@(x,y) power(atan(x.*y),3)+power(sin(x),2);
star = 2*pi;
x = 0:0.01:star;  % define range and mesh of x and y which will be shown in figure
y = -1:0.01:1;
[X, Y] = meshgrid(x, y);
figure
surf(X, Y, t(X,Y))
f = power(atan(x1*y1),3)+power(sin(x1),2);
f2 = diff(f,x1) == 0;
f3 = diff(f,y1) == 0;
[sx1,sy1] = vpasolve(f2,f3);%find the values of x and y of the minimum
extreme_values = subs(f, {x1,y1}, {sx1,sy1})%solve for the minimum
%}
%
%Q3
eqn = (x2^2)+2*x2+(y2^2)+2*y2*(z2^2)+(z2^2);
eqn1 = diff(eqn,x2)==0
eqn2 = diff(eqn,y2)==0
eqn3 = diff(eqn,z2)==0
[sx,sy,sz] = vpasolve(eqn1,eqn2,eqn3) %find the inflection points
vari = {x2, y2, z2};
eqn1ma = [diff(eqn,x2) diff(eqn,y2) diff(eqn,z2)];
eqn2ma = [a a a; a a a; a a a];
eqn2sol = zeros(3,3);
for i = 1:3
    for j=1:3
        eqn2ma(i,j)=diff(eqn1ma(1,j),vari(i));%create Hessian
    end
end
eqn2ma
lambdia = [lamb 0 0; 0 lamb 0; 0 0 lamb];
eqn2malamb = eqn2ma-lambdia;
lamb0 = det(eqn2malamb);
S = solve(lamb0, lamb)
eigen = eig(eqn2ma)
for i = 1:3
    for j = 1:3
        for k = 1:3
            eqn2sol(j,k)=subs(eqn2ma(j,k), {x2,y2,z2}, {sx(i,1),sy(i,1),sz(i,1)});
        end
    end
    eqn2sol %ouput Hessian
    eigenvalules = eig(eqn2sol)%find eigenvalue
end

Could you tell me why?

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Steven Lord 2023-1-31

Which two quantities are you comparing and expecting to be the same? Please add semicolons to all the lines of code that lack them then at the end display the two quantities that you expected to be equal.

Torsten 2023-1-31

If you subs the {sx,sy,sz} in the symbolic expression "eigen", you will get the same numerical eigenvalues as you get from the line "eigenvalules = eig(eqn2sol)".

So I don't understand your point.

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Answer 1

Sulaymon Eshkabilov 2023-1-31

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1904050-why-my-symbolic-result-of-eigenvalue-doesn-t-match-my-numeric-result#answer_1160725

在 MATLAB Online 中打开

The reason for having three different eigen values can be explained with the followings:

Eigenvalues1 = eig([2 0 0; 0 2 0; 0 0 2]) % @ (-1   0   0)
Eigenvalues1 = 3×1
     2
     2
     2
Eigenvalues2 = eig([2 0 0; 0 2 4*-sqrt(.5); 0 4*-sqrt(.5) 4*-.5+2]) % @ (-1 -.5  -sqrt(.5))
Eigenvalues2 = 3×1
   -2.0000
    2.0000
    4.0000
Eigenvalues3 = eig([2 0 0; 0 2 4*sqrt(.5); 0 4*sqrt(.5) 4*-.5+2])   % @ (-1 -.5   sqrt(.5))
Eigenvalues3 = 3×1
   -2.0000
    2.0000
    4.0000

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Why my symbolic result of eigenvalue doesn't match my numeric result?

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Why my symbolic result of eigenvalue doesn't match my numeric result?

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