Collapse a tensor

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.

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: $$ E_{k_1 \ldots k_d}=X_{i_1\ldots i_n j_1\ldots j_n k_1 \ldots k_d}\delta_{i_1j_1}\ldots \delta_{i_nj_n}{ E_{k_1...k_d}=X_{i_1...i_n j_1...j_n k_1 ... k_d}\delta_{i_1j_1} ... \delta_{i_nj_n } }$$

See also

mul.tensor, to.tensor

Author

K. Gerald van den Boogaart

Examples

A <- to.tensor(1:20,c(i=2,j=2,k=5))
A
#> , , 1
#> 
#>       j
#> i      [,1] [,2]
#>   [1,]    1    3
#>   [2,]    2    4
#> 
#> , , 2
#> 
#>       j
#> i      [,1] [,2]
#>   [1,]    5    7
#>   [2,]    6    8
#> 
#> , , 3
#> 
#>       j
#> i      [,1] [,2]
#>   [1,]    9   11
#>   [2,]   10   12
#> 
#> , , 4
#> 
#>       j
#> i      [,1] [,2]
#>   [1,]   13   15
#>   [2,]   14   16
#> 
#> , , 5
#> 
#>       j
#> i      [,1] [,2]
#>   [1,]   17   19
#>   [2,]   18   20
#> 
#> attr(,"class")
#> [1] "tensor"
trace.tensor(A,"i","j")
#> [1]  5 13 21 29 37
#> attr(,"class")
#> [1] "tensor"