Degeneracy (mathematics) Totally Explained

Everything about Degenerate Math totally explained

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's 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.

Degenerate rectangle

For any non-empty subset

S

of the indices

= [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).

Further Information

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