rankcondition

Rank condition is a criterion expressed in terms of the rank of a matrix or linear map that guarantees a property of a system, a map, or a model.

In linear algebra, for a linear system Ax = b, the rank condition states that the system is consistent if rank([A|b]) = rank(A). It has a unique solution when rank(A) equals the number of variables, and it has infinitely many solutions when rank(A) < number of variables and the system is consistent.

In control theory, rank conditions determine dynamical properties such as controllability and observability. The Kalman rank condition says that the pair (A, B) is controllable if the controllability matrix [B AB A^2B ... A^{n-1}B] has full rank n. The dual condition applies to observability with (C, A).

In differential geometry and algebraic geometry, rank refers to the derivative or a morphism between vector bundles. The rank theorem describes local normal forms near a point where the Jacobian has rank r. Determinantal varieties are loci where a map has rank at most r, defined by the vanishing of all (r+1)×(r+1) minors.

Examples: a 2×2 matrix has rank ≤ 1 exactly when its determinant is zero; a 3×2 matrix has rank ≤ 1 if all its 2×2 minors vanish.

See also: rank, Jacobian, determinantal variety, Kalman criterion.