Editing Ellipsoid (section)

=== As a quadric ===
If {{mvar|'''A'''}} is a real, symmetric, {{mvar|n}}-by-{{mvar|n}} [[positive-definite matrix]], and {{mvar|'''v'''}} is a vector in <math>\R^n,</math> then the set of points {{math|'''x'''}} that satisfy the equation
:<math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math>
is an ''n''-dimensional ellipsoid centered at {{mvar|'''v'''}}. The expression <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) </math> is also called the '''ellipsoidal norm''' of {{math|'''x''' − '''v'''}}. For every ellipsoid, there are unique {{mvar|'''A'''}} and {{math|'''v'''}} that satisfy the above equation.<ref name=":0" />{{Rp|page=67|location=}}

The [[eigenvector]]s of {{mvar|'''A'''}} are the principal axes of the ellipsoid, and the [[eigenvalue]]s of {{mvar|'''A'''}} are the reciprocals of the squares of the semi-axes (in three dimensions these are {{math|''a''<sup>−2</sup>}}, {{math|''b''<sup>−2</sup>}} and {{math|''c''<sup>−2</sup>}}).<ref>{{cite web |url=http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |title=Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD |access-date=2013-10-12 |url-status=live |archive-url=https://web.archive.org/web/20130626233838/http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |archive-date=2013-06-26}} pp. 17–18.</ref> In particular:

* The [[diameter]] of the ellipsoid is twice the longest semi-axis, which is twice the square-root of the reciprocal of the largest eigenvalue of {{mvar|'''A'''}}.
* The [[width]] of the ellipsoid is twice the shortest semi-axis, which is twice the square-root of the reciprocal of the smallest eigenvalue of {{mvar|'''A'''}}.

An invertible [[linear transformation]] applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable [[rotation]], a consequence of the [[polar decomposition]] (also, see [[spectral theorem]]). If the linear transformation is represented by a [[symmetric matrix|symmetric 3&nbsp;×&nbsp;3 matrix]], then the eigenvectors of the matrix are orthogonal (due to the [[spectral theorem]]) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The [[singular value decomposition]] and [[polar decomposition]] are matrix decompositions closely related to these geometric observations.

For every positive definite matrix <math>\boldsymbol{A}</math>, there exists a unique positive definite matrix denoted {{math|'''''A'''''<sup>1/2</sup>}}, such that  <math>\boldsymbol{A} = \boldsymbol{A}^{1/ 2}\boldsymbol{A}^{1/ 2}; </math> this notation  is motivated by the fact that this matrix can be seen as the "positive square root" of <math>\boldsymbol{A}.</math> The ellipsoid defined by <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math> can also be presented as<ref name=":0">{{Cite Geometric Algorithms and Combinatorial Optimization}}</ref>{{Rp|page=67|location=}}<blockquote><math>A^{-1/2}\cdot S(\mathbf{0},1) + \mathbf{v}</math></blockquote>where S('''0''',1) is the [[unit sphere]] around the origin.