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 degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation.

Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an nth degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiply coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.

In quantum mechanics any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels. Usually any such degeneracy indicates some underlying symmetry in the system.

Degenerate rectangle

For any non-empty subset $S \subseteq \{1, 2, \ldots, n\}$, there is a bounded, axis-aligned degenerate rectangle

$R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\}$

where $\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]$ 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).