= Canonical form of a quadric
{title2=$\lambda_1X_1^2+\cdots+\lambda_nX_n^2+\ell^TX+c=0$}
The equation of a <quadric surface> after <orthogonal diagonalization> of its <symmetric matrix> and appropriate <translation>. The <spectral theorem for real symmetric matrices> removes mixed quadratic terms, and completing squares removes linear terms in directions with nonzero <eigenvalues>. Linear terms in zero-<eigenvalue> directions can remain and generate <elliptic paraboloids> or <hyperbolic paraboloids>. The signs, zero coefficients and constant distinguish ordinary surfaces, <degenerate conics> in plane sections, and empty real loci.
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