Double dot product vs double inner product Anything involving tensors has 47 different names and notations, and I am having trouble getting any consistency out of it.
This document (http://www.polymerprocessing.com/notes/root92a.pdf) clearly ascribes to the colon symbol (as "double dot product"):
$\mathbf{T}:\mathbf{U}=T_{ij} U_{ji}$
while this document (http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html) clearly ascribes to the colon symbol (as "double inner product"):
$\mathbf{T}:\mathbf{U}=T_{ij} U_{ij}$
Same symbol, two different definitions.  To make matters worse, my textbook has:
$\mathbf{\epsilon}:\mathbf{T}$
where $\epsilon$ is the Levi-Civita symbol $\epsilon_{ijk}$ so who knows what that expression is supposed to represent.
Sorry for the rant/crankiness, but it's late, and I'm trying to study for a test which is apparently full of contradictions.  Any help is greatly appreciated.

I know this is old, but this is the first thing that comes up when you search for double inner product and I think this will be a helpful answer for others.
The operation $\mathbf{A}*\mathbf{B} = \sum_{ij}A_{ij}B_{ji}$ is not an inner product because it is not positive definite. Calling it a double-dot product is a bit of a misnomer. Meanwhile, for real matricies, $\mathbf{A}:\mathbf{B} = \sum_{ij}A_{ij}B_{ij}$ is the Frobenius inner product. They can be better realized as,
$$\mathbf{A}:\mathbf{B} = \operatorname{tr}\left(\mathbf{A}\mathbf{B}^\mathsf{T}\right) $$
$$\mathbf{A}*\mathbf{B} = \operatorname{tr}\left(\mathbf{A}\mathbf{B}\right) $$
This definition for the Frobenius inner product comes from that of the dot product, since for vectors $\mathbf{a}$ and $\mathbf{b}$,
$$\mathbf{a}\cdot\mathbf{b} = \operatorname{tr}\left(\mathbf{a}\mathbf{b}^\mathsf{T}\right)$$
We then can even understand how to extend this to complex matricies naturally by the vector definition. Since for complex vectors, we need the inner product between them to be positive definite, we have to choose,
$$(\mathbf{a},\mathbf{b}) = \mathbf{a}\cdot\overline{\mathbf{b}}^\mathsf{T} = a_i \overline{b}_i$$
as our inner product. Likewise for the matrix inner product, we have to choose,
$$\mathbf{A}:\mathbf{B} = \operatorname{tr}\left(\mathbf{A}\mathbf{B}^\mathsf{H}\right) = \sum_{ij}A_{ij}\overline{B}_{ij}$$
where $\mathsf{H}$ is the conjugate transpose operator.
As for the Levi-Cevita symbol, the symmetry of the symbol means that it does not matter which way you perform the inner product.
