Trace/Matrix/Linear mapping/Introduction/Section

Definition

Let ${}K$ be a field and let ${}M={\left(a_{ij}\right)}_{ij}$ be an ${}n\times n$-matrix over ${}K$. Then

${}\operatorname {trace} {\left(M\right)}:=\sum _{i=1}^{n}a_{ii}\,$

is called the trace of ${}M$.

Definition

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let ${}\varphi \colon V\rightarrow V$ be a linear mapping, which is described by the matrix ${}M$ with respect to a basis. Then ${}\operatorname {trace} {\left(M\right)}$ is called the trace

of ${}\varphi$, written as ${}\operatorname {trace} {\left(\varphi \right)}$.

Because of exercise, this is independent of the basis chosen. The trace is a linear form on the vector space of all square matrices, and on the vector space of all endomorphisms.