On the rank of m×2×2 and m×3×2 tensors over arbitrary fields

Abstract

In this paper, we provide exact rank computations for $m\times 2\times 2$ and $m\times 3\times 2$ tensors over arbitrary fields. By analyzing the structural properties of slice matrices, we reduce the tensor rank problem to computations involving matrix ranks and diagonalizations. This yields a complete and explicit rank classification for these families of tensors and provides a clearer structural understanding on rank of small tensors.