Direct Sum and Tensor Product (Outer Product)

[Eddie W. Shore]

DIRSUM: Direct Sum

DIRSUM is the direct sum of two tensors, specifically column vectors. The direct sum, symbolized by ⊕ (circle with a plus symbol in it), stacks column vectors on top of each other. The order of the two vectors matters.

Example:

V1 = [ [ 2 ] [ 3 ] ]
V2 = [ [ 5 ] [ 6 ] [ 8 ] ]

V1 ⊕ V2 = [ [ 2 ] [ 3 ] [ 5 ] [ 6 ] [ 8 ] ]

V2 ⊕ V1 = [ [ 5 ] [ 6 ] [ 8 ] [ 2 ] [ 3 ] ]

The dimension of the direct sum is the sum of the dimensions of the vectors.


TENSOR: Tensor Product

The tensor product, also known as the outer product multiplies the numbers from V1 to each element V2 in order. The tensor product can be represented in a column vector or a matrix.

If V1 and V2 are matrices, the outer product is calculated as:

V1 ⊗ V2 = V1 × V2ᵀ = V1 × transpose(V2)

Example:

V1 = [ [ 2 ] [ 3 ] ]
V2 = [ [ 5 ] [ 6 ] [ 8 ] ]

V1 ⊕ V2 = [ [ 10, 12, 16 ] [ 15, 18, 24 ] ]

V2 ⊕ V1 = [ [ 10, 15 ] [ 12, 18 ] [ 16, 24 ] ]

The tensor product is not commutative, order matters.

Code:

EXPORT DIRSUM(v,w)
BEGIN
// direct sum of 2 vectors
// 2023-04-30 EWS
LOCAL lv,lw,lc;
lv:=mat2list(v); 
lw:=mat2list(w);
lc:=CONCAT(lv,lw);
RETURN list2mat(lc,1);
END;

Code:

EXPORT TENSOR(v,w)
BEGIN
// tensor product of 2 vectors or matrices
// 2023-04-23 EWS
RETURN v*TRN(w);
END;

Source:

Bradley, Tai-Danae. "The Tensor Products, Demystified" math3ma.com November 18, 2018. https://www.math3ma.com/blog/the-tensor-...emystified Accessed April 24, 2023.