Degeneracy (mathematics)
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In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class.
- A point is a degenerate circle, namely one with radius 0. The circle is a degenerate form of an ellipse, namely one with eccentricity 0.
- The line is a degenerate form of a parabola if the parabola resides on a tangent plane. Also it is a degenerate form of a rectangle, if this has a side of length 0.
- A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.
- A set containing a single point is a degenerate continuum.
- See "general position" for other examples.
Another usage of the word comes in eigenproblems: a degenerate eigenvalue is one that has more than one linearly independent eigenvector.
Degenerate rectangle
For any non-empty subset of the indices <math>\{1, 2, ..., n\},</math> a bounded degenerate rectangle <math>R</math> is a subset of <math>\mathcal{R}^n</math> of the following form:
<math>R = \left\{\mathbf{x} : x_i = c_i \ (\mathrm{for} \ i\in S) \ \mathrm{and} \ a_i \leq x_i \leq b \ (\mathrm{for} \ i \notin S)\right\}</math>
where <math>\mathbf{x}= [x_1, x_2, \ldots, x_n]</math>. The number of degenerate sides of <math>R</math> is the number of elements of the subset <math>S</math>. Thus, there may be as few as one degenerate "side" or as many as <math>n</math> (in which case <math>R</math> reduces to a singleton point).
See also: degeneracy, Trivial (mathematics).