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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.
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 matrix vectorization operation can be written in terms of a linear sum. Let X be an m × n matrix that we want to vectorize, and let e i be the i -th canonical basis vector for the n -dimensional space, that is e i = [ 0 , … , 0 , 1 , 0 , … , 0 ] T {\textstyle \mathbf {e} _{i}=\left[0,\dots ,0,1,0,\dots ,0\right]^{\mathrm {T} }} .
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
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 first line of the article says "the trace of an n-by-n diagonal matrix A is defined to be the sum of the elements ...", whereas, the first example shows calculation of trace for a non-diagonal matrix. As far as I know, trace is defined for square matrix, and does not require diagonal matrix. -- Samikrc 05:30, 7 February 2011 (UTC)