Spårmängderna

Spårmängderna, a term used in mathematical literature, refer to the sets of trace values obtained from elements of a given subset of matrices or linear operators. Given a subset A of the matrix algebra End(V) over a field F (for example, A ⊆ M_n(F)), the trace set of A is defined as Tr(A) = {tr(a) : a ∈ A}, where tr denotes the usual trace.

Basic properties and construction

The trace is a linear functional, so Tr(A) respects addition and scalar multiplication in the sense that Tr(a + b) = tr(a) + tr(b) and Tr(ca) = c tr(a) for a ∈ A and c ∈ F, provided the operations are defined within the chosen subset. The trace is invariant under similarity: tr(PAP⁻¹) = tr(A) for any invertible P. Consequently, all matrices within a similarity class share the same trace, and Tr on a set invariant under conjugation is determined by a single representative.

Special cases

If A is the entire matrix algebra M_n(F), then Tr(A) = F when F is infinite, since values of tr(A) can span the field (e.g., tr(αI) = αn). If A consists of strictly upper triangular matrices, every element has trace zero, so Tr(A) = {0}. For diagonal matrices, the trace equals the sum of diagonal entries, illustrating how Tr(A) reflects aggregate diagonal information of the elements.

Applications and context

Spårmängderna are a convenient way to summarize the range of trace values for a family of operators. They connect to representation theory as character values, to operator algebras through traces on algebras, and to linear algebra as a tool for studying invariant properties under conjugation.

See also: trace, trace-class, linear functional, conjugacy, representation theory.