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  "body": "\n## Summary\n\nTwo square matrices $A$ and $B$ are congruent when $B = P^{T} A P$ for some invertible matrix $P$.\nCongruent matrices are the matrices of one [[bilinear form|0SQZMXB6GP]] in two different bases.\nCongruence is an equivalence relation, and its classification for real symmetric matrices is [[Sylvester's law of inertia|0WFED3K3DQ]].\nIt differs from [[similarity|0E89HQATM8]], which uses $P^{-1}$ in place of $P^{T}$.\n\n## Explanations\n\n### L1 human\n\nA bilinear form, such as a dot product, gives a matrix once you choose a basis.\nIf you change the basis with a matrix $P$, the new matrix of the same form is $P^{T} A P$.\nSo congruent matrices describe the same form seen in two coordinate systems.\nFor example, $\\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $\\begin{pmatrix} 4 \u0026 0 \\\\ 0 \u0026 9 \\end{pmatrix}$ are congruent, with $P = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 3 \\end{pmatrix}$.\n",
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