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:

Notation:
The trace of a matrix $A$ is also commonly denoted as $\operatorname{Tr}(A)$ or $\operatorname{Tr}A$ .

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 \begin{eqnarray*} \operatorname{trace}(A+B) &=& \operatorname{trace}(A)+ \operatorname{trace}(B), \\ \operatorname{trace}(cA) &=& c\cdot \operatorname{trace}(A). \end{eqnarray*}
2. For the transpose and conjugate transpose, we have for any square matrix $A$ with real (or complex) entries, \begin{eqnarray*} \operatorname{trace} (A^t) &=& \operatorname{trace} (A), \\ \operatorname{trace} (A^\ast) &=& \overline{\operatorname{trace} (A)}. \end{eqnarray*}
3. If $A$ and $B$ are matrices such that $AB$ is a square matrix, then $$ \operatorname{trace} (AB) = \operatorname{trace} (BA).$$

   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 $$ \operatorname{trace} (ABC) = \operatorname{trace} (CAB) = \operatorname{trace} (BCA).$$

4. If $B$ is in invertible square matrix of same order as $A$ , then $$ \operatorname{trace} (A) = \operatorname{trace} (B^{-1}A B).$$ 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 \begin{eqnarray*} \operatorname{trace} A^\ast A &=& \operatorname{trace} A A^\ast \\ &=& \sum_{i,j=1}^n |a_{ij}|^2. \end{eqnarray*}Here $^\ast$ is the complex conjugate, and $|\cdot|$ is the complex modulus. In particular, $\operatorname{trace} A^\ast A\ge 0$ with equality if and only if $A=0$ . (See the Frobenius matrix norm.)
6. Various inequalities for $\operatorname{trace}$ are given in [2].

See the proof of properties of trace of a matrix.

Bibliography

1

The Trace of a Square Matrix. Paul Ehrlich, [online] http://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, online.