trace

The trace $\operatorname{Tr}(A)$ of a square matrix $A$ is defined to be the sum of the diagonal entries of $A$ . It satisfies the following formulas:

•

$\operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B)$
•

$\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ ()

where $A$ and $B$ are square matrices of the same size.

The trace $\operatorname{Tr}(T)$ of a linear transformation $T\colon V\longrightarrow V$ from any finite dimensional vector space $V$ to itself is defined to be the trace of any matrix representation of $T$ with respect to a basis of $V$ . This scalar is independent of the choice of basis of $V$ , and in fact is equal to the sum of the eigenvalues of $T$ (over a splitting field of the characteristic polynomial), including multiplicities.

The following link presents some examples for calculating the trace of a matrix.

A trace on a $C^{*}$ -algebra $A$ is a positive linear functional $\phi\colon A\to\mathbb{C}$ that has the .

Title	trace
Canonical name	Trace
Date of creation	2013-03-22 12:17:57
Last modified on	2013-03-22 12:17:57
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	10
Author	mhale (572)
Entry type	Definition
Classification	msc 15A15
Classification	msc 15A04
Related topic	FrobeniusMatrixNorm