yawiki.org entry for Degeneracy (mathematics)

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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.

    Degeneracy (mathematics)
        Degenerate rectangle
        See also

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Degenerate rectangle

For any non-empty subset

S

of the indices

,

a bounded degenerate rectangle

R

is a subset of

mathcal^n

of the following form:

R = left

where

mathbf= 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).

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