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  "body": "\n## Definition\n\nLet $F$ be a field, let $d \\ge 1$, and let $V_1, \\ldots, V_d$ be finite-dimensional [[vector spaces|000AYPA9AJ]] over $F$.\nA *simple tensor* in the [[tensor product|0BF2DE2SYK]] $V_1 \\otimes \\cdots \\otimes V_d$ is a nonzero tensor of the form $v_1 \\otimes \\cdots \\otimes v_d$ with $v_i \\in V_i$.\nThe *rank* of a tensor $T \\in V_1 \\otimes \\cdots \\otimes V_d$ is the smallest integer $r \\ge 0$ such that $T$ is a sum of $r$ simple tensors.\nThe zero tensor has rank $0$.\n\n## Remarks\n\nFor $d = 2$ and $V_1 = F^m$, $V_2 = F^n$, a tensor is an $m \\times n$ matrix, a simple tensor is a rank-one matrix $u v^T$, and the tensor rank is the [[rank of the matrix|0ZC9YTXZRY]].\nExample: in $F^2 \\otimes F^2 \\otimes F^2$ with standard basis $e_1, e_2$, the tensor $T = e_1 \\otimes e_1 \\otimes e_1 + e_2 \\otimes e_2 \\otimes e_2$ has rank $2$.\nThe written sum shows rank at most $2$.\nThe tensor is not simple: flattening it to a $2 \\times 4$ matrix gives rank $2$, while a simple tensor flattens to a matrix of rank $1$.\nNon-example of rank $2$: over $\\C$, the tensor $e_1 \\otimes e_1 \\otimes e_2 + e_1 \\otimes e_2 \\otimes e_1 + e_2 \\otimes e_1 \\otimes e_1$ has rank $3$, although it is a limit of tensors of rank $2$.\nSo the set of tensors of rank at most $r$ need not be closed, which leads to the notion of border rank.\nThe rank of a real tensor can be larger than its rank over $\\C$.\nOther names are canonical polyadic rank and CP rank.\n",
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