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Direct Sum and Tensor Product (Outer Product) - Eddie W. Shore - 05-04-2023 07:56 PM
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-product-demystified Accessed April 24, 2023.