Inner Rank and Lower Bounds for Matrix Multiplication

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$ matrix multiplication over an arbitrary field is at least $2n^2-n+1$. While inner rank does not provide improvements to currently known lower bounds, we argue that this notion merits further study.