trace of a matrix

Definition
Let $A=(a_{i,j})$ be a square matrix oforder $n$.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 tothe 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 anysquare 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 }$

   $=$${\displaystyle \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.