= Level sets of a positive semidefinite quadratic form
{title2=$x^TAx=E,\quad A\geq0$}
For a real <symmetric matrix> $A$ with nonnegative <eigenvalues>, the <finite-dimensional spectral theorem> changes the equation to $\sum_i\lambda_iq_i^2=E$ in an <orthonormal eigenbasis>. If $E>0$ and $A$ has positive rank, the positive-eigenvalue coordinates form an <ellipsoid>, while the zero-eigenvalue coordinates are arbitrary. The level set is therefore a cylinder over that <ellipsoid>, with axis directions given by the <kernel of a linear map> of $A$. For rank two in three dimensions, it is an <elliptic cylinder>. At $E=0$ the level set is exactly the <kernel of a linear map>; for $E<0$ it is empty. This distinguishes a <positive semidefinite matrix> from a <positive-definite matrix>, whose positive level sets are bounded <ellipsoids>.
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