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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $n \\ge 1$.\nLet $A$ and $B$ be $n \\times n$ matrices over $F$.\nThen $A$ and $B$ are *congruent* if there is an [[invertible|0EB9N0S33C]] $n \\times n$ matrix $P$ over $F$ with\n$$B = P^{T} A P.$$\nHere $P^{T}$ is the [[transpose|09DXWYRD43]] of $P$.\n\n## Remarks\n\nExample: over $\\R$, let $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ and $P = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 -1 \\end{pmatrix}$.\nThen $P$ is invertible, since $\\det P = -2$, and $P^{T} A P = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 -2 \\end{pmatrix}$, so these two matrices are congruent.\nNon-example: over $\\R$, the identity $I_2$ and $-I_2$ are not congruent, because $x^{T} P^{T} I_2 P x = \\norm{Px}^2 \\ge 0$ for all $x \\in \\R^2$, while $x^{T} (-I_2) x \u003c 0$ for $x \\ne 0$.\nCongruence is an equivalence relation on $n \\times n$ matrices.\nLet $A$ be the matrix of a [[bilinear form|0SQZMXB6GP]] in one basis, and let $P$ be the [[change of basis matrix|0MKRHZB3WS]] to a second basis.\nThen $P^{T} A P$ is the matrix of the same form in the second basis; see [[change of basis for a bilinear form|0SPYAGE7T0]].\nCongruence keeps symmetry and the rank, and it multiplies the determinant by the nonzero square $(\\det P)^2$.\nOver $\\C$, the complex analogue often uses the [[conjugate transpose|0DXKE10PHF]] $P^{*}$ in place of $P^{T}$; this is called $*$-congruence.\n",
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