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.

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Degenerate rectangle

See also

Degenerate rectangle

For any non-empty subset $S$ of the indices $\{1,\; 2,\; ...,\; n\},$ a bounded degenerate rectangle $R$ is a subset of $mathcal\{R\}^n$ of the following form:
$R\; =\; left\{mathbf\{x\}\; :\; x\_i\; =\; c\_i\; (mathrm\{for\}\; iin\; S)\; mathrm\{and\}\; a\_i\; leq\; x\_i\; leq\; b\; (mathrm\{for\}\; i$
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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

★ Trivial (mathematics)