Degeneracy (mathematics)
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This article is about degeneracy in mathematics. For other uses, see Degeneracy.
For the degeneracy of a graph, see Arboricity#Related_concepts.
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.
- A segment 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.
- A random variable which can only take one value has a degenerate distribution.
- 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.
[edit] Degenerate rectangle
For any non-empty subset 
, there is a bounded, axis-aligned degenerate rectangle
where ![\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]](http://upload.wikimedia.org/math/d/c/0/dc08451f67a834356992f61420320021.png)
and ai,bi,ci are constant (with 
for all i). The number of degenerate sides of R is the number of elements of the subset S. Thus, there may be as few as one degenerate "side" or as many as n (in which case R reduces to a singleton point).
