Degeneracy (mathematics)
From Wikipedia, the free encyclopedia
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.
[edit] Degenerate rectangle
For any non-empty subset S of the indices {1,2,...,n}, a bounded degenerate rectangle R is a subset of 
of the following form:
where ![\mathbf{x}= [x_1, x_2, \ldots, x_n]](../../../math/a/d/b/adb05d00944414b0326983377d24ffa7.png)
. 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).
