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The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed points of the corresponding permutation, because the diagonal term a ii is 1 if the i th point is fixed and 0 otherwise. The trace of a projection matrix is the dimension of the target space.
A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). 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.
The trace formula has applications to arithmetic geometry and number theory.For instance, using the trace theorem, Eichler and Shimura calculated the Hasse–Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula.
The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.
Trace formula may refer to: Arthur–Selberg trace formula , also known as invariant trace formula, Jacquet's relative trace formula, simple trace formula, stable trace formula Grothendieck trace formula , an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology , used to express the Hasse–Weil zeta function .
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.
When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to Tr L/K (xy) is a nondegenerate, symmetric bilinear form called the trace form. If L/K is a Galois extension, the trace form is invariant with respect to the Galois group.
The Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology. One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups.