determinantach

Determinantach is a fictional or speculative invariant associated with square matrices, introduced in thought experiments and nonstandard mathematical discussions to explore how different ways of aggregating eigenvalues affect matrix properties. It is not a standard concept in linear algebra, and its precise use varies by author or context.

Definition. For an n by n matrix A over a field F, let λ1, …, λn be the eigenvalues of A in an algebraic closure. Choose a fixed function f: F → F. The determinantach of A with respect to f, denoted DetAch_f(A), is defined as the product DetAch_f(A) = ∏_{i=1}^n f(λi). Because it is built from the eigenvalues, DetAch_f(A) is invariant under similarity transformations (since similarity preserves the spectrum).

Special cases and related ideas. If f is the identity function f(λ) = λ, DetAch_f(A) equals the ordinary determinant. If f(λ) = λ^m for a positive integer m, DetAch_f(A) equals det(A)^m. Choosing other functions f yields a family of related invariants that depend on the spectrum but not on the full Jordan form in general. The construction can be used to illustrate how different spectral aggregations interact with matrix operations such as direct sums (DetAch_f(A ⊕ B) = DetAch_f(A) DetAch_f(B)).

Limitations and context. Because determinantach depends on a chosen function f, its information content and utility are context-dependent and inherently not unique. In standard linear algebra, the determinant and characteristic polynomial remain the primary spectral-related invariants. Determinantach is typically discussed as a playful or didactic device to compare how varying f reshapes invariants.

See also. Determinant, eigenvalues, spectrum, characteristic polynomial, matrix invariants.