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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.

An elliptic cone is a real quadric surface consisting of the two nappes $u=\pm \sqrt{(b{v}^2+c{w}^2)/a}$, joined at the vertex. Its intersections with planes $u=u_0\ne 0$ are ellipses. It is the zero level of an indefinite quadratic form of signature $(1,2)$. Changing the level to a positive value produces a two-sheeted hyperboloid; changing it to a negative value produces a one-sheet hyperboloid, when the positive coefficient is assigned to the axial coordinate $u$.

For $a,b>0$, an elliptic cylinder is a quadric surface with the displayed equation in suitable Cartesian coordinates, independent of the third coordinate. It is formed by translating an ellipse parallel to a line perpendicular to its plane. The generator direction is the zero-eigenvalue direction of its quadratic part, while the transverse semiaxes are $a,b$. It is circular if $a=b$.

For $a,b>0$, this quadric surface is independent of its third coordinate and is formed by translating a hyperbola along that direction. Its two connected components correspond to $u\ge a$ and $u\le -a$. In elliptic cylindrical coordinates, fixing a generic angular coordinate selects a half-branch of such a cylinder because the signs of both planar coordinates are fixed by that angular value.

An orthogonal change of coordinates relates two orthonormal bases of a real inner product space. With the new basis vectors as columns of an orthogonal matrix $Q$, old and new coordinate columns obey $x=Qy$. Then $x^{T}x=y^{T}y$, so Euclidean norm, Euclidean distance and angles are preserved. A quadratic form $x^{T}Ax$ becomes $y^T(Q^{T}AQ)y$. For a symmetric matrix, orthogonal diagonalization chooses $Q$ so that the transformed quadratic form is a sum of eigenvalues times squared coordinates. This explains the principal-axis descriptions of a quadric surface.

A quadric surface obtained by extending a parabola along a fixed direction outside its plane. For example, $x=c-y^2/(4c)$ with arbitrary $z$ is a parabolic cylinder parallel to the $z$ axis. Each cross-section of fixed $z$ has vertex $(c,0)$ and focus $(0,0)$ in its $xy$ plane. It occurs as a coordinate surface in parabolic cylindrical coordinates.

The quadric surface $x^2+y^2+z^2=1$ is the unit sphere centered at the origin. Each section $z=c$, for $|c|<1$, is a circle of radius $\sqrt{1-c^2}$.

The quadric surface $z=x^2+y^2$ is a circular elliptic paraboloid, opening in the positive $z$ direction with vertex at the origin. Its horizontal sections $z=c>0$ are circles of radius $\sqrt{c}$, and its vertical sections through the $z$ axis are parabolas.

For $a{x}^2+y^2=0$, $z$ is unrestricted. If $a>0$, both nonnegative terms must vanish, so the locus is the $z$ axis, a straight line. If $a=0$, it is the plane $y=0$, represented by the repeated factor $y^2$. If $a<0$, factorization givesso the locus is two distinct planes, $y=\pm \sqrt{-a}x$, meeting along the $z$ axis. These are degenerate quadrics, not ordinary curved cylinders.

A complex $3\times 3$ matrix is diagonalizable exactly when it has a basis of three linearly independent eigenvectors. If all its eigenvalues are one, a diagonalization would have $D=I$ and hence $A=PI{P}^{-1}=I$. Therefore the specified matrix, which is not a scalar identity matrix, is not diagonalizable.

For the displayed real symmetric matrix, choose the orthogonal matrixIts columns form an orthonormal eigenbasis, andIn the orthogonal coordinates $u=(x+y)/\sqrt{2}$, $v=(x-y)/\sqrt{2}$, the quadric surface has equation $u^2+3v^2+3z^2=1$. It is a prolate spheroid, with long semiaxis one in direction $(1,1,0)$ and two equal short semiaxes $1/\sqrt{3}$. The figure gives its principal-axis sketch.

A diagonalizable matrix can certainly have exactly two distinct eigenvalues: $\mathrm{diag}(1,1,2)$ is already diagonal and has an eigenbasis despite its repeated eigenvalue.

Use the orthogonal change of coordinatesIt preserves distances and angles and gives $x^{T}Mx=x^{\prime T}D{x}^{\prime }$. Thus the quadric surface is expressed along its principal axes asThe following three cases are the subparts of (ii) in the original PDF.

In the orthonormal basis of part (b), the three quadric surfaces have the following forms. For $\alpha =1$,a two-sheeted hyperboloid with axis $u$, vertices $(\pm 1/\sqrt{6},0,0)$, and one sheet on each side of the plane $u=0$. For $\alpha =0$,an elliptic cone with vertex at the origin and the same axis. For $\alpha =-1$,a one-sheet hyperboloid with axis $u$ and a waist ellipse at $u=0$, whose semiaxes are $1/\sqrt{3}$ along $v$ and $1/2$ along $w$. The sections at fixed $u$ enlarge away from the waist.

The orthogonal change of coordinates preserves distance to the origin. On the required quadric surface,Thus the squared distance is at least $1/4$. Equality holds precisely when $u=v=0$, and the constraint then gives $w=\pm 1/2$. Using $e_w=(1,0,-3)/\sqrt{10}$ from part (b), the nearest points in the original coordinates areSubstitution checks the constraint directly: $5x^2-3z^2+6xz=(5-27-18)/40=-1$. The inequality proves these are global nearest points, rather than merely stationary candidates.

After an orthogonal diagonalization, the sign of the third coefficient classifies the displayed quadric surface. If $C>0$, it is an ellipsoid with semiaxes $A^{-1/2},B^{-1/2},C^{-1/2}$. If $C=0$, it is an elliptic cylinder parallel to the $w$ axis, with elliptical transverse section. If $C<0$, it is a one-sheet hyperboloid with axis $w$ and waist ellipse $A{u}^2+B{v}^2=1$. The continuous limit $C\to 0$ sends the associated axial length scale to infinity. Repeated positive eigenvalues make the corresponding principal-axis choices nonunique within their eigenspace, without changing the surface.