# Trace   - properties and formulas -

Definition
- Definition
- Examples

Prpoperties
- Symmetry
- Cyclic property
- Linearity
Definition
The trace of an $n \times n$ square matrix $A$ is the sum of the diagonal components of the matrix,
where $A_{ij}$ is $i$-th row and $j$-th column element of $A$.
Examples
Let $A$ be a squared matrix defined as
The trace of $A$ is
Let $B$ be a squared matrix defined as
The trace of $A$ is
Symmetry
The trace of the product of two matrices is equal to the trace of the product in which order is swapped. Let $A$ and $B$ be an $m \times n$ and an $n \times m$ matrix, respectively, then

Proof
$AB$ is an $m \times m$ matrix. By the definition of trace, the trace of $AB$ is
Using the rule of matrix product, we have
then

Cyclic property
Let $A$, $B$, $C$ be $m\times n$, $n \times l$, $l \times m$ matrix, respectively. The trace of the product of three matrices is invariant even if the order of the products is cyclically changed:
In other words, the trace is invariant under cyclic permutations
This property is called cyclic property of the trace.
Proof
We use the symmetry of the trace. Using the symmetry of $A$ and $BC$, we have
Using the symmetry of $AB$ and $C$, we have
Therefore,

Linearity
Let $A$ and $B$ be square matrices and $\alpha$ and $\beta$ be scalars. The trace is linear:

Proof
By the definition of trace, we see that