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matlabeigenvalueeigenvector

Spectral decomposition of Hessian

I am running into issues when attempting to shift a Hessian to be positive definite for an optimization problem in Matlab. An example of my problem is:

H=[1 2 2; 2 3 2; 1 3 1]; [V,D]=eig(H);

While H*V-V*D, as it should, essentially equals zero:

V*D*V' does not provide the original H matrix

Solution

  • I actually tried to run your code:

    H = [
  1 2 2; 
  2 3 2;
  1 3 1
];

[V,D] = eig(H);

test = norm(H*V-V*D,inf) / norm(V*D,inf);

    and I'm seeing no obvious problems with the example you posted. The eigenvalues and vectors that I get satisfy their defining equation with high accuracy:

    test = 7.57596318689868e-16

    On the top of that, as @kpg987 pointed out, you have to use the inverse of V, not the transposed version of V. And if you perform the following test:

    test = V * D * inv(V);

    You will obtain your original hessian (or something very very close to it).