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Tensor Direct Sum

The tensor direct sum of two tensors is their block-diagonal combination on the direct sums of their corresponding factor vector spaces. If T in V_1 tensor ... tensor V_d and U in W_1 tensor ... tensor W_d, then T direct sum U belongs to (V_1 direct sum W_1) tensor ... tensor (V_d direct sum W_d) and has zero entries in every mixed block (Li 2026). For order 2, this is the matrix direct sum.

Concatenating decompositions gives

r(T direct sum U)<=r(T)+r(U),

where r is tensor decomposition rank. Equality must not be assumed for arbitrary tensors. Li (2026) established additivity of both tensor decomposition rank and symmetric tensor rank for two and three copies of a particular Comon conjecture counterexample.

See also

Comon Conjecture, Matrix Direct Sum, Tensor Decomposition Rank, Tensor Direct Product

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References

Li, J. "The Comon Rank Gap of a Third-Order Symmetric Tensor Can Exceed One." 1 Oct 2026. https://arxiv.org/abs/2610.01660.

Cite this as:

Weisstein, Eric W. "Tensor Direct Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TensorDirectSum.html

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