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Given a function :, the associated trace function on is given by = (), where has eigenvalues and stands for a trace of the operator. Convexity and monotonicity of the trace function [ edit ]
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.
The trace operator can be defined for functions in the Sobolev spaces , with <, see the section below for possible extensions of the trace to other spaces. Let Ω ⊂ R n {\textstyle \Omega \subset \mathbb {R} ^{n}} for n ∈ N {\textstyle n\in \mathbb {N} } be a bounded domain with Lipschitz boundary.
A consequence of the Baker–Campbell–Hausdorff formula is the following result about the trace: = + . That is to say, since each z j {\displaystyle z_{j}} with j ≥ 2 {\displaystyle j\geq 2} is expressible as a linear combination of commutators, the trace of each such terms is zero.
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 physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by Golden (1965) and Thompson (1965). It has been developed in the context of statistical mechanics , where it has come to have a particular significance.
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Von Neumann's trace inequality See also: List of things named after John von Neumann: Mathematics, Quantum mechanics: John von Neumann: Weinberg–Witten theorem: Quantum Gravity: Steven Weinberg and Edward Witten: Weyl character formula See also: List of things named after Hermann Weyl: Mathematics: Hermann Weyl: Wien's law: Physics: Wilhelm Wien