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  "body": "\n## Statement\n\nLet $A$ be a real symmetric $n \\times n$ matrix.\nLet $P$ and $P'$ be invertible real $n \\times n$ matrices such that $D = P^T A P$ and $D' = P'^T A P'$ are diagonal.\nThen $D$ and $D'$ have the same number of positive diagonal entries and the same number of negative diagonal entries.\nThey also have the same number of zero diagonal entries.\n\n## Remarks\n\nSo these three numbers are invariants of the quadratic form $x \\mapsto x^T A x$; they give its [[signature|07EZA80NEH]].\nThey are also the numbers of positive, negative and zero eigenvalues of $A$, counted with multiplicity.\nEquivalently, two real symmetric matrices are congruent if and only if they have the same numbers of positive, negative and zero eigenvalues.\n\n## History\n\nJames Joseph Sylvester published the theorem in 1852.\n\n## Explanations\n\n### L1 human\n\nA real quadratic form can be written as a sum of squares in many ways.\nThe numbers of plus signs and minus signs are always the same.\nFor example, $x^2 + 2xy$ always has one plus sign and one minus sign.\n",
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