Mineralogy
"Hardness" is determined by comparison with other minerals. In the Mohs scale, a standard set of minerals are numbered in order of increasing hardness from 1 (talc) to 10 (diamond). A harder mineral will scratch a softer, so an unknown mineral can be placed in this scale by which minerals it scratches and which scratch it. A few minerals such as calcite and kyanite have a hardness that depends significantly on direction. Hardness can also be measured on an absolute scale using a sclerometer; compared to the absolute scale, the Mohs scale is nonlinear.
Stiffness matrix
By defining the vector "F" with components "F" = formula_5, the coefficients "u" are determined by the linear system "AU" = "F". The stiffness matrix is symmetric, i.e. "A" = "A", so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that the system "AU" = "F" always has a unique solution. (For other problems, these nice properties will be lost.)