determinant1

Determinant1 is a directional variant of the determinant used in a hypothetical or pedagogical setting to quantify how a linear map scales a chosen one-dimensional subspace of a vector space. For an n×n matrix A over a field F and a fixed 1-dimensional subspace V spanned by a nonzero vector v, determinant1(A,V) is defined when V is invariant under A, meaning A v ∈ V. In that case A induces a scalar action on V, and determinant1(A,V) is that scalar, equivalently the eigenvalue of A associated with V. If V is not invariant, some authors discuss extensions, but standard practice uses only invariant subspaces.

Computationally, determinant1 reduces to finding an eigenvector for A corresponding to V and taking the corresponding eigenvalue. In a basis where V is spanned by the first basis vector e1 and A has a block form with a scalar λ in the (1,1) position, determinant1(A,V) = λ. This differs from the ordinary determinant, which multiplies all invariant scales across all directions.

Examples: For A = [[2,1],[0,3]] and V = span{(1,0)}, determinant1(A,V) = 2. For a diagonal A = diag(4,5,6) and V = span{e2}, determinant1(A,V) = 5.

Applications of determinant1 are primarily pedagogical: it helps relate the concept of volume scaling to a specific direction and clarifies when a full determinant cannot be reduced to a single scalar on a chosen subspace. Limitations include dependence on the chosen subspace and the requirement that V be A-invariant. See also determinant, eigenvalues, invariant subspaces. Determinant1 is presented here as a hypothetical construct to illustrate directional invariants and is not a standard mathematical concept.