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 $\backslash \{1,\; 2,\; ...,\; n\backslash \},$ a bounded degenerate rectangle $R$ is a subset of $\backslash mathcal\{R\}^n$ of the following form:

$R\; =\; \backslash left\backslash \{\backslash mathbf\{x\}\; :\; x\_i\; =\; c\_i\; \backslash \; (\backslash mathrm\{for\}\; \backslash \; i\backslash in\; S)\; \backslash \; \backslash mathrm\{and\}\; \backslash \; a\_i\; \backslash leq\; x\_i\; \backslash leq\; b\; \backslash \; (\backslash mathrm\{for\}\; \backslash \; i\; \backslash notin\; S)\backslash right\backslash \}$

where $\backslash mathbf\{x\}=x\_2,\; \backslash ldots,\; x\_n$. 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).

See also: degeneracy, Trivial (mathematics).

Elementary mathematics