Mathematical objects called tensors can be used to represent multidimensional objects. In reality a scalar is rank 0 tensor, so scalar is the simplest tensor. Rank of a tensor can be thought as objects spatial dimension. Let's consider matrix multiplication, where there are two indices, tensor of rank 2. If we have two matrices $A_{ij}$ and $B_{ij}$. We can multiply them with index notation as follows using an obscure language R with a naive algorithm:

> set.seed(42) > A <- matrix(rnorm(4), 2, 2) > B <- matrix(rnorm(4), 2, 2) > C <- matrix(0, 2, 2) > for(i in 1:2) { + for(j in 1:2) { + for(k in 1:2) { + C[i,j] = C[i,j] + A[i, k] * B[k,j] + } + } + } > C [,1] [,2] [1,] 6.413566 -2.5178879 [2,] -2.249789 0.4908884 > A %*% B [,1] [,2] [1,] 0.5156982 2.0378605 [2,] -0.2954518 -0.9134599

You should have noticed that we introduce a third index. That is called a dummy index which we actually do the sum. This approach is technically called Einstein summation rule. So we represent the multiplication as follows: $C_{ij} = \sum_{k=1}^{m} A_{ik} B_{kj}$. Normally sum is dropped for short notation. It is quite intuitive from this notation that number of columns of A should be equal to number of rows in B to perform the sum consistently.

There are software packages that can do symbolic and numerical tensor algebra . Here we demonstrate an R package called tensorA developed by Professor van den Boogaart of TU Freiberg. We can repeat the matrix multiplication by using this package as follows.

> library("tensorA") > set.seed(42) > At <- to.tensor(rnorm(4), c(i=2, k=2)) > Bt <- to.tensor(rnorm(4), c(k=2, j=2)) > At %e% Bt j i [,1] [,2] [1,] 0.5156982 2.0378605 [2,] -0.2954518 -0.9134599 attr(,"class") [1] "tensor" "matrix"

Note that here $\%e\%$ implies tensor multiplication with Einstein summation rule. This is pretty trivial example but if you imagine higher dimensional objects, tracking indices would be cumbersome so multi-linear algebra makes life easy.

In conclusion, I think, using tensor arithmetic for multidimensional arrays looks more compacts and efficient (~ 2-3 times). A discussion related to this appeared in R help list.

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