Rank additivity for matrix-multiplication tensors

Determine whether rank is additive for direct sums of matrix-multiplication tensors; specifically, establish whether the rank of a direct sum of independent matrix-multiplication tensors always equals the sum of the individual ranks.

Background

The paper discusses direct sums of independent matrix products and notes that rank additivity, although conjectured by Strassen, is false for general tensors. The corresponding question for matrix-multiplication tensors remains unresolved. Establishing additivity would characterize the exact rank cost of computing independent matrix products separately versus jointly.

References

Rank additivity of matrix-multiplication remains an open problem.

— Simultaneous Matrix Multiplication by Tensor Contraction Aggregation  (2610.26563 - Solomonik et al., 19 Oct 2026) in Section 2.1, “Direct sums of independent matrix–matrix products”