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The trace of an n × n square matrix A is defined as [1] [2] [3]: 34 = = = + + + where a ii denotes the entry on the i th row and i th column of A. The entries of A can be real numbers , complex numbers , or more generally elements of a field F .
a labeling V 2 → Hom(V,V) associating each degree-2 vertex to a linear transformation. Note that V 2 and V n should be considered as distinct sets in the case n = 2. A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices V 1 into two disjoint ordered collections called the inputs and the outputs.
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra.
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.
If L/K is separable then each root appears only once [2] (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K ] times 1). More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, [1] i.e.,
Weak trace class operators. Since Com(L 1,∞) + = (L 1) + the co-dimension of the commutator subspace of the weak-L 1 ideal is infinite. Every trace on weak trace class operators vanishes on trace class operators, and hence is singular. The weak trace class operators form the smallest ideal where every trace on the ideal must be singular. [18]
In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi.The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free ...
str(T) = the ordinary trace of T 00 − the ordinary trace of T 11. Let us show that the supertrace does not depend on a basis. Suppose e 1, ..., e p are the even basis vectors and e p+1, ..., e p+q are the odd basis vectors. Then, the components of T, which are elements of A, are defined as