trace of a matrix
Definition
Let $A=({a}_{i,j})$ be a square matrix^{} of
order $n$.
The trace of the matrix is the sum of the main diagonal:
$\mathrm{trace}(A)=\sum _{i=1}^{n}{a}_{i,i}$
Notation:
The trace of a matrix $A$ is also commonly denoted as $\mathrm{Tr}(A)$
or $\mathrm{Tr}A$.
Properties:

1.
The trace is a linear transformation from the space of square matrices to the real numbers. In other words, if $A$ and $B$ are square matrices with real (or complex) entries, of same order and $c$ is a scalar, then
$\mathrm{trace}(A+B)$ $=$ $\mathrm{trace}(A)+\mathrm{trace}(B),$ $\mathrm{trace}(cA)$ $=$ $c\cdot \mathrm{trace}(A).$ 
2.
For the transpose^{} and conjugate transpose^{}, we have for any square matrix $A$ with real (or complex) entries,
$\mathrm{trace}({A}^{t})$ $=$ $\mathrm{trace}(A),$ $\mathrm{trace}({A}^{\ast})$ $=$ $\overline{\mathrm{trace}(A)}.$ 
3.
If $A$ and $B$ are matrices such that $AB$ is a square matrix, then
$$\mathrm{trace}(AB)=\mathrm{trace}(BA).$$ For this reason it is possible to define the trace of a linear transformation, as the choice of basis does not affect the trace. Thus, if $A,B,C$ are matrices such that $ABC$ is a square matrix, then
$$\mathrm{trace}(ABC)=\mathrm{trace}(CAB)=\mathrm{trace}(BCA).$$ 
4.
If $B$ is in invertible^{} square matrix of same order as $A$, then
$$\mathrm{trace}(A)=\mathrm{trace}({B}^{1}AB).$$ In other words, the trace of similar matrices^{} are equal.

5.
Let $A$ be a square matrix of order $n$ with real (or complex) entries ${a}_{ij}$. Then
$\mathrm{trace}{A}^{\ast}A$ $=$ $\mathrm{trace}A{A}^{\ast}$ $=$ $\sum _{i,j=1}^{n}}{{a}_{ij}}^{2}.$ Here ${}^{\ast}$ is the complex conjugate^{}, and $\cdot $ is the complex modulus^{}. In particular, $\mathrm{trace}{A}^{\ast}A\ge 0$ with equality if and only if $A=0$. (See the Frobenius matrix norm.)

6.
Various inequalities for $\mathrm{trace}$ are given in [2].
See the proof of properties of trace of a matrix.
References
 1 The Trace of a Square Matrix. Paul Ehrlich, [online] http://www.math.ufl.edu/ ehrlich/trace.htmlhttp://www.math.ufl.edu/ ehrlich/trace.html
 2 Z.P. Yang, X.X. Feng, A note on the trace inequality for products of Hermitian matrix^{} power, Journal of Inequalities in Pure and Applied Mathematics, Volume 3, Issue 5, 2002, Article 78, http://www.emis.de/journals/JIPAM/v3n5/082_02.htmlonline.
Title  trace of a matrix 

Canonical name  TraceOfAMatrix 
Date of creation  20130322 11:59:56 
Last modified on  20130322 11:59:56 
Owner  Daume (40) 
Last modified by  Daume (40) 
Numerical id  20 
Author  Daume (40) 
Entry type  Definition 
Classification  msc 15A99 
Related topic  ShursInequality 