Definition | |
---|---|

- | Definition |

- | Examples |

Properties | |
---|---|

- | Symmetry |

- | Cyclic property |

- | Linearity |

- | Defined by three properties |

Definition

The trace of an $n \times n$ square matrix $A$ is the sum of the diagonal components of the matrix,
Examples

Let $A$ be a squared matrix defined as
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
$AB$ is an $m \times m$ matrix. By the definition of trace, the trace of $AB$ is

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:
We use the symmetry of the trace. Using the symmetry of $A$ and $BC$, we have

Linearity

Let $A$ and $B$ be square matrices and
$\alpha$ and $\beta$ be scalars.
The trace is linear:
Defined by three properties

Let $A$ and $B$ are $n \times n$ matrices,
$\alpha$ and $\beta$ be scalars,
and $f$ be a function of $n \times n$ matrix.
If the function $f$ satisfies
Matrix $E_{ij}$

Let $E_ {ij}$
be $n \times n$ matrix with only the
$i$-th row and
$j$-th column element being $1$
and the other components being $0$.
Proof

From
Prop.$(1)$
and
Eq.$(6.2)$,