extensordominant

Extensordominant is a term encountered in some mathematical discussions to describe a property of linear operators with respect to exterior powers. It is not a widely standardized notion, and its meaning can vary across sources. The term draws on the idea of an extensor (a k-vector in the exterior algebra) and a dominance condition observed in the operator’s action on exterior powers.

In a finite-dimensional vector space V, let A be a linear operator A: V → V. For each k = 1, ..., dim V, A induces the map Λ^k A: Λ^k V → Λ^k V on the k-th exterior power. A is said to be extensordominant if, for every k, the induced map Λ^k A has a unique eigenvalue of maximal modulus and this eigenvalue corresponds to the exterior product of the k most expanding directions of A. Equivalently, in each exterior power the spectral radius is achieved by a single eigenvalue, which dominates the rest.

A practical sufficient condition often cited is that A is diagonalisable with eigenvalues ordered by decreasing modulus, with strict inequalities between successive moduli. Under such a condition, Λ^k A exhibits a unique top eigenvalue for each k, yielding a uniform sense of dominance across exterior powers. The property is preserved under powers of A and under change of basis.

Examples and caveats: A diagonal matrix diag(3, 2, 1) is extensordominant in the above sense, since the eigenvalues in each exterior power come from products of the largest moduli and are uniquely maximal. If eigenvalues have equal moduli or occur in complex conjugate pairs with equal magnitude, the dominance may fail, depending on the chosen interpretation.

See also: exterior algebra, extensor, dominant eigenvalue, Perron–Frobenius theory.