Collapses the tensor over dimensions i and j. This is like a trace for
matrices or like an inner product of the dimensions i and j.
Usage
trace.tensor(X,i,j)
Arguments
X
the tensor
i
a numeric or character vector of dimensions of X,
used for the inner product.
j
a numeric or character vector of dimensions of X with
the same length but other elements than i.
Value
A tensor like X with the i and j dimensions removed.
Details
Let be
$$X_{i_1\ldots i_n j_1\ldots j_n k_1 \ldots k_d}$$ the tensor. Then the result is given by
$$E_{k_1 \ldots k_d}=sum_{i_1\ldots i_n} X_{i_1\ldots i_n
i_1\ldots i_n k_1 \ldots k_d} $$
With the Einstein summing convention we would write:
$$
E_{k_1 \ldots k_d}=X_{i_1\ldots i_n j_1\ldots j_n k_1 \ldots
k_d}\delta_{i_1j_1}\ldots \delta_{i_nj_n}{
E_{k_1...k_d}=X_{i_1...i_n j_1...j_n k_1 ...
k_d}\delta_{i_1j_1} ... \delta_{i_nj_n
}
}$$