Computing eigenspectrum of a Riemannian Hessian matrix

[Rajendra]

Dear sir,

I have two doubts,

1. I get different eigenspectrum of  a Riemannian Hessian matrix when I use the 'hessianspectrum(problem, X)' function and use eigs function on the matrix getHessian(problem, X, Z), where X is a point on the manifold and the point of the convergence of the RTR method and Z is a tangent vector in the tangent space at the point X.

2. How to analytically prove  that a Riemannian Hessian matrix (Hess f(X) [Z] ) is a positive definite matrix or not? Will the definiteness of the Riemannian Hessian matrix depend on the choice of the tangent vector Z at the point X on the manifold?

with regards,

Rajendra

[BM]

Hello Rajendra, 

Thanks for the questions. 

1. I get different eigenspectrum of  a Riemannian Hessian matrix when I use the 'hessianspectrum(problem, X)' function and use eigs function on the matrix getHessian(problem, X, Z), where X is a point on the manifold and the point of the convergence of the RTR method and Z is a tangent vector in the tangent space at the point X.

I should mention that

getHessian(problem, X, Z) is not the Hessian, but it is "Hessian along Z". For example, if X were a vector (so, Z is a vector too), then getHessian(problem, X, Z) is "Hessian at X multiplied with Z". Hence, doing eigs on getHessian(problem, X, Z) does not make sense. 

The correct spectrum is indeed given by hessianspectrum(problem, X). 

2. How to analytically prove  that a Riemannian Hessian matrix (Hess f(X) [Z] ) is a positive definite matrix or not? Will the definiteness of the Riemannian Hessian matrix depend on the choice of the tangent vector Z at the point X on the manifold?

No, positiveness does not depend on Z. We look at hessianspectrum(problem, X) and see whether they are positive. 

Regards,

Bamdev