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  "body": "\n## Definition\n\nLet $R$ be a ring and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe matrix $A$ is *symmetric* when $A^T = A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.\nEquivalently, $a_{ij} = a_{ji}$ for all $i$ and $j$.\n\n## Remarks\n\nThe matrix $\\begin{pmatrix} 1 \u0026 7 \\\\ 7 \u0026 2 \\end{pmatrix}$ is symmetric.\nFor each $m \\times n$ matrix $B$ over a commutative ring, $B^T B$ and $B B^T$ are symmetric.\nA real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, by the [[spectral theorem|06J4PXRG22]].\nSymmetric matrices represent [[symmetric bilinear forms|0AXZNE5XNH]] in a basis.\n\n## Explanations\n\n### L1 human\n\nA symmetric matrix looks the same after we flip it over its main diagonal.\nThe entry in row $i$, column $j$ equals the entry in row $j$, column $i$.\nDistance tables between cities are symmetric, because the distance from $A$ to $B$ equals the distance from $B$ to $A$.\n",
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