What is the colon operator between matrices?

Question

While reading Robust Quasistatic Finite Elements and Flesh Simulation by Teran et al., I have seen in several equations a colon operator used between matrices.

Here are some examples, using matrices $F$, $P$ and $C$:

$\delta F : \delta P > 0$

$\delta F : (\partial P / \partial F) : \delta F > 0$

$i2 = C : C$

The only hint I have is that I believe that $C$ is a diagonal matrix with diagonal elements $[\sigma_1^2, \sigma_2^2, \sigma_3^2]$, and the result of $C : C$ is $\sigma_1^4 + \sigma_2^4 + \sigma_3^4$.

Does anybody know what this operator represents?

Answer 1

Since the paper deals with tensors etc, I think it's the "double dot product" as described here:

https://en.wikipedia.org/wiki/Dyadics

Double dot product

$$ A:B = \sum_{j} \sum_{i} (a_i\cdot d_j)(b_i\cdot c_j), $$ or $$ A:B = \sum_{j} \sum_{i} (a_i\cdot c_j)(b_i\cdot d_j) $$

where $A = \sum_{i} a_i b_i$ and $B = \sum_j c_j d_j$.