The tensor product of two arrays is notionally an
outer product of the arrays collapsed in specific extents by
summing along the appropriate diagonals. For example, a matrix
product is the tensor product along the second extent of the
first matrix and the first extent of the second. Thus
%*% B could also be evaluated as
tensor(A, B, 2, 1),
A %*% t(B) could be
tensor(A, B, 2, 2).
Numerical vectors, matrices or arrays
This code does the ‘obvious’ thing, which is to perm the
"along" extents to the end (for
A) and beginning (for
B) of the two objects and then do a matrix multiplication
Generally, an array with dimension comprising the
remaining extents of
A concatenated with the remaining
B are completely collapsed then the
result is a scalar without a
dim attribute. This
is quite deliberate and consistent with the general rule that the
dimension of the result is the sum of the original dimensions
less the sum of the collapse dimensions (and so could be zero).
A 1D array of length 1 arises in a different set of
circumstances, eg if
A is a 1 by 5 matrix and
a 5-vector then
tensor(A, B, 2, 1) is a 1D array of length
Some special cases of
tensor may be
independently useful, and these have got shortcuts as follows.
|%*t%|| Matrix product
|%t*%|| Matrix product
|%t*t%|| Matrix product
Jonathan Rougier, [email protected]
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A <- matrix(1:6, 2, 3) dimnames(A) <- list(happy = LETTERS[1:2], sad = NULL) B <- matrix(1:12, 4, 3) stopifnot(A %*% t(B) == tensor(A, B, 2, 2)) A <- A %o% A C <- tensor(A, B, 2, 2) stopifnot(all(dim(C) == c(2, 2, 3, 4))) D <- tensor(C, B, c(4, 3), c(1, 2)) stopifnot(all(dim(D) == c(2, 2))) E <- matrix(9:12, 2, 2) s <- tensor(D, E, 1:2, 1:2) stopifnot(s == sum(D * E), is.null(dim(s)))
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