= Stationary directions of the Rayleigh quotient
{title2=$\nabla R_A(x)=\dfrac{2}{x^Tx}(Ax-R_A(x)x)$}
For a real <symmetric matrix> $A$ and nonzero real <vector> $x$, the <quotient rule> and the <quadratic form gradient> give the displayed <gradient>. It vanishes exactly when $Ax=R_A(x)x$, so the <stationary points> are <eigenvectors>. Equivalently, the <Lagrange multiplier> condition for optimizing $x^TAx$ on the <unit sphere> is $Ax=\lambda x$. The smallest and largest <eigenvalues> give the global extrema; other <eigenvectors> can be stationary without being extrema, as a direction mixing larger and smaller <eigenvalues> changes the quotient in opposite ways.
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