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  "body": "\n## Summary\n\nThe rank of a tensor is the smallest number of simple tensors $v_1 \\otimes \\cdots \\otimes v_d$ whose sum is the tensor.\nFor tensors of order $2$, that is matrices, it is the usual [[rank of a matrix|0ZC9YTXZRY]].\nFor order $3$ and higher it is much harder to understand: it can depend on the field, and computing it is NP-hard.\nThe rank of the matrix multiplication tensor governs the [[matrix multiplication exponent|X0G02CGQ5J]].\n\n## Explanations\n\n### L1 human\n\nA simple tensor is the most basic kind of tensor, an outer product of vectors, like a rank-one matrix $u v^T$.\nThe rank counts how many such building blocks are needed to write a given tensor.\nA matrix of rank $r$ is a sum of $r$ rank-one matrices and no fewer.\nFor a three-way array, the tensor $e_1 \\otimes e_1 \\otimes e_1 + e_2 \\otimes e_2 \\otimes e_2$ needs two blocks, so its rank is $2$.\n",
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