You can check for the related discussion in the following book on Kronecker product and Matrix Calculus by Prof. Alexander Graham, Dover Publications.

An effective approach depends on the character of the matrices. For example… In solving the non-ideal Gibbs problem (chemical reactions involving substances that are not ideal gases), the Hessian (partial derivatives of the Gibbs free energy with respect to molar abundances) becomes increasingly ill-conditioned as the species depart from ideality (either compressibility less than one or fugacity coefficient less than one). This means that the steepest-descent method fails. A practical solution evaded chemists for many years. I stumbled across the solution 30 years ago while working on a totally different problem. What I'm suggesting is that there may be a solution to your particular problem but it may be very difficult to find. I was lucky. If I saw the matrices, I might be able to come up with something.

Dudley, hi: Thanks for the detailed response. This is the so-called Christoffel symmetric matrix (appears in the anisotropic elasticity) minus the identity matrix. The matrix is small, 3x3, and the explicit expressions for each component are known. Assume all components depend on three Cartesian coordinates. The determinant of the matrix always vanishes. I need to find the derivatives and second derivatives (the gradient and the Hessian) of this vanishing determinant wrt the Cartesian coordinates. The inverse of the matrix does not exist

I have solved problems with anisotropic elasticity using FEM and FDM. There may be a way of side-stepping this problem by taking a different approach altogether.

Won't the Euler Formula work in some way for testing homogeneity?

Sir S.K.Mallick

for RHMHM School