Degeneracy (mathematics)

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This article is about degeneracy in mathematics. For other uses, see Degeneracy.

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.
A random variable which can only take one value is degenerate.
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, axis-aligned degenerate rectangle $R$ is a subset of $\mathcal{R}^n$ of the following form:

$R = \left\{\mathbf{x} : x_i = c_i \ (\mathrm{for} \ i\in S) \ \mathrm{and} \ a_i \leq x_i \leq b_i \ (\mathrm{for} \ i \notin S)\right\}$

where $\mathbf{x}= [x_1, x_2, \ldots, x_n]$ and $a i, b i, c i$ are constant (with $a_i \leq b_i$ 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).