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If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic.
The trace, Tr L/K (α), is defined as the trace (in the linear algebra sense) of this linear transformation. [ 1 ] For α in L , let σ 1 ( α ), ..., σ n ( α ) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K ).
The right hand side shows the resulting 2-by-2 reduced density matrix . In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar-valued function on operators, the partial trace is an operator-valued function.
An n-trace diagram is a graph = (,), where the sets V i (i = 1, 2, n) are composed of vertices of degree i, together with the following additional structures: a ciliation at each vertex in the graph, which is an explicit ordering of the adjacent edges at that vertex;
Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of , evaluated at the identity matrix, is equal to the trace.The differential ′ is a linear operator that maps an n × n matrix to a real number.
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.
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 determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues. [4] [5] [6] From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this ...