|
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. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation. == Formal definition == Let ''V'' be a vector space of dimension ''n'' over a field ''F'' (with ''n''≥2), and let Fun(''V'',''V'') denote the linear transformations on ''V''. 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; * a labeling ''V''2 → Fun(''V'',''V'') associating each degree-2 vertex to a linear transformation. Note that ''V''2 and ''Vn'' should be considered as distinct sets in the case ''n'' = 2. A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices ''V''1 into two disjoint ordered collections called the ''inputs'' and the ''outputs''. The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph: * Loops are permitted (a loop is an edges that connects a vertex to itself). * Edges that have no vertices are permitted, and are represented by small circles. * Multiple edges between the same two vertices are permitted. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Trace diagram」の詳細全文を読む スポンサード リンク
|