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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.
A trace diagram representing the adjugate of a matrix. In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix.
The partial trace is performed over a subsystem of 2-by-2 dimension (single qubit density matrix). 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.
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.
For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling. American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators .
One can show that the trace-norm is a norm on the space of all trace class operators () and that (), with the trace-norm, becomes a Banach space. When H {\displaystyle H} is finite-dimensional, every (positive) operator is trace class and this definition of trace of A {\displaystyle A} coincides with the definition of the trace of a matrix .
The trace of a rotation matrix is equal to the sum of its eigenvalues. For n = 2, a rotation by angle θ has trace 2 cos θ. For n = 3, a rotation around any axis by angle θ has trace 1 + 2 cos θ. For n = 4, and the trace is 2(cos θ + cos φ), which becomes 4 cos θ for an isoclinic rotation.
Let be the -subalgebra of the matrix algebra () generated by the preimages of elements of . The algebra A {\displaystyle A} is then as simple as possible, more precisely: [ 3 ] If Γ {\displaystyle \Gamma } is of the first or second type then A {\displaystyle A} is a quaternion algebra over k {\displaystyle k} .