What does the notation $A:B$ means for matrices $A$ and $B$?

Question

What does the notation $A:B$ means for matrices $A$ and $B$? I saw this is an equation derived from the Navier-Stokes equation. For example,

\begin{equation*} \int_{\Omega}\nabla(u):\nabla(v)dx \end{equation*}

for vector fields $v$ and $u$.

Answer 1

This might be a double-dot product, which unfortunately is used in two different conventions:

$$A:B=\sum_i\sum_j(a_i\cdot d_j)(b_i\cdot c_j)$$ $$A:B=\sum_i\sum_j(a_i\cdot c_j)(b_i\cdot d_j)$$ where $A=\sum_i a_ib_i$ and $B=\sum_i c_id_i$ are two general dyadics.

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In line with uranix's comment, if you let $A=\nabla u$ and $B=\nabla v$, you get $A:B=\Delta(u\cdot v)$, in the second convention at least.