detAdetB

detAdetB is a theoretical construct in abstract algebra, defined as a joint determinant-like invariant associated with a pair of n×n matrices A and B over a field F. It is designed to capture how two determinant-like quantities interact under simultaneous similarity transformations, and to provide a single scalar that reflects certain joint properties of the pair.

Definition: Let A, B ∈ F^{n×n}. The detAdetB invariant is defined by detAdetB(A,B) = det(A) det(B) + Tr(adj(A) B). Here adj(A) denotes the adjugate of A. This quantity remains unchanged when A and B are simultaneously conjugated by the same invertible matrix P, i.e., detAdetB(P^{-1}AP, P^{-1}BP) = detAdetB(A,B).

Properties: It is polynomial in the entries of A and B, and linear in the adjugate term. It is generally not symmetric in A and B, but it is invariant under joint changes of basis.

Example: Take A = [[1,0],[0,1]] and B = [[2,0],[0,3]]. Then det(A) = 1, det(B) = 6, adj(A) = I, Tr(adj(A) B) = Tr(B) = 5, so detAdetB(A,B) = 16 + 5 = 11.

Applications and context: detAdetB is discussed in theoretical treatments of matrix pair invariants and can be used to distinguish certain equivalence classes of matrix pairs under simultaneous similarity. It has potential as a compact descriptor in systems theory and in exploratory linear algebra research.

Origin: DetAdetB appears in discussions of determinant-adjacent invariants and is not part of standard linear algebra practice. It is described in speculative or didactic contexts to illustrate how determinant-like terms can be combined.

See also: determinant, adjugate, matrix invariant, simultaneous similarity.