Degeneracy (mathematics)

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

$R=\left\{\mathbf {x} :x_{i}=c_{i}\ (\mathrm {for} \ i\in S)\ \mathrm {and} \ a_{i}\leq x_{i}\leq b\ (\mathrm {for} \ i\notin S)\right\}$

where $\mathbf {x} =[x_{1},x_{2},\ldots ,x_{n}]$. 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)