Tensor products of $k$-tensors

tensorprod(U, ...)
tensorprod2(U1,U2)

Arguments

U,U1,U2: Object of class ktensor
...: Further arguments, currently ignored

Details

Given a $k$-tensor $S$ and an $l$-tensor $T$, we can form the tensor product $S\otimes T$, defined as

$$S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)= S\left(v_1,\ldots v_k\right)\cdot T\left(v_{k+1},\ldots v_{k+l}\right).$$

Package idiom for this includes tensorprod(S,T) and S %X% T; note that the tensor product is not commutative. Function tensorprod() can take any number of arguments (the result is well-defined because the tensor product is associative); it uses tensorprod2() as a low-level helper function.

Value

The functions documented here all return a spray object.

References

Spivak 1961

Note

The binary form %X% uses uppercase X to avoid clashing with %x% which is the Kronecker product in base R.

Author

Robin K. S. Hankin

Examples


(A <- ktensor(spray(matrix(c(1,1,2,2,3,3),2,3,byrow=TRUE),1:2)))
#> A linear map from V^3 to R with V=R^3:
#>            val
#>  2 3 3  =    2
#>  1 1 2  =    1
(B <- ktensor(spray(10+matrix(4:9,3,2),5:7)))
#> A linear map from V^2 to R with V=R^19:
#>            val
#>  16 19  =    7
#>  15 18  =    6
#>  14 17  =    5
tensorprod(A,B)
#> A linear map from V^5 to R with V=R^19:
#>                  val
#>  1 1 2 14 17  =    5
#>  2 3 3 14 17  =   10
#>  1 1 2 15 18  =    6
#>  2 3 3 15 18  =   12
#>  1 1 2 16 19  =    7
#>  2 3 3 16 19  =   14

A %X% B - B %X% A
#> A linear map from V^5 to R with V=R^19:
#>                    val
#>   2  3 3 16 19  =   14
#>   1  1 2 16 19  =    7
#>   2  3 3 15 18  =   12
#>  16 19 1  1  2  =   -7
#>   1  1 2 15 18  =    6
#>   2  3 3 14 17  =   10
#>   1  1 2 14 17  =    5
#>  16 19 2  3  3  =  -14
#>  15 18 2  3  3  =  -12
#>  14 17 2  3  3  =  -10
#>  15 18 1  1  2  =   -6
#>  14 17 1  1  2  =   -5


Va <- matrix(rnorm(9),3,3)
Vb <- matrix(rnorm(38),19,2)

LHS <- as.function(A %X% B)(cbind(rbind(Va,matrix(0,19-3,3)),Vb))
RHS <-  as.function(A)(Va) * as.function(B)(Vb)

c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
#>       LHS       RHS      diff 
#> -0.582478 -0.582478  0.000000

Tensor products of \(k\)-tensors