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  "body": "\n## Definition\n\nLet $R$ be a commutative ring and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe *trace* of $A$ is\n$$\\tr A = \\sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \\cdots + a_{nn}.$$\n\n## Remarks\n\nThe trace of $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$ is $5$.\nThe trace is [[linear|0GJWQZCPZM]], it is [[invariant under transpose|09E1RCZ7YT]], and [[$\\tr(AB) = \\tr(BA)$|0Y4NF1DRKD]].\nSimilar matrices have the same trace, so the [[trace of a linear map|0KZDNGA561]] is well defined.\nOver an algebraically closed field, the trace is the [[sum of the eigenvalues|0NMXDCNNKR]], counted with multiplicity.\n\n## Explanations\n\n### L1 human\n\nThe trace adds up the numbers on the main diagonal of a square matrix.\nIt is a single number that does not change when we change the basis.\nFor the identity matrix of order $n$, the trace is $n$.\n",
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