I have been told that the relation $$\vec v /k_d=\vec v \cdot \overline{\overline{k^{-1}}} \cdot \vec v$$ states that the scalar $k_d$ is a weighted harmonic average of the eigenvalues of $\overline{\overline{k}}$.
Here, $\vec v$ and $\overline{\overline{k}}$ are respectively first and second order (rank) tensors in $\mathbb{R}^3$.
I have a few questions. First, is this statement true? I thought that for something to be an average (a mean) it would be a sum of a collection of numbers divided by the number of numbers in the collection. In the relation above I do not see where the number of numbers in the collection are included...
Second, I am curious if the relation above is equivalent to the following: $$\frac{v_i^2}{k_d}=k_{ij}^{-1} v_i v_j$$ $$k_d = \frac{v_i^2}{k_{ij}^{-1} v_i v_j}$$ $$\frac{1}{k_d}=\frac{\cos^2 \beta_x}{k_x}+\frac{\cos^2 \beta_y}{k_y}+\frac{\cos^2 \beta_z}{k_z}$$ where the $k_i$ are the principal values of $k_{ij}$ and $\beta_i$ are the angles between the vector $v_i$ and the three principal axes, $x,y,z$, respectively. Note, I'm not sure if I am correctly employing the summation convention rules in the above, so please correct me if wrong.