📂Matrix Algebra

Trace

Definition

Let matrix $n\times n$ be given as follows.

$$ A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} $$

The sum of the diagonal elements of $A$ is defined as the trace of $A$ and is denoted as follows.

$$ \text{tr}(A)=\text{Tr}(A)=a_{11}+a_{22}+\cdots + a_{nn}=\sum \limits_{i=1}^{n} a_{ii} $$

Explanation

The trace can also be thought of as a function as follows. Let $M_{n\times n}(\mathbb{R})$ be the set of $n\times n$ matrices with real number components. Then, $\text{Tr}$ is defined as the following function.

$$ \text{Tr} : M_{n\times n} (\mathbb{R}) \to \mathbb{R},\quad \text{Tr}(A)=\sum \limits_{i=1}^{n} a_{ii} $$

Properties

Let $A,B,C$ be a $n \times n$ matrix, and let $k$ be a constant.

(a) The trace of a scalar multiple is the same as the scalar multiple of the trace.

$$ \text{Tr}(kA)= k\text{Tr}(A) $$

(b) The trace of a sum is the same as the sum of the traces.

$$ \text{Tr}(A+B)=\text{Tr}(A)+\text{Tr}(B) $$

(a)+(b) The trace is linear.

$$ \text{Tr}(kA+B)=k\text{Tr}(A)+\text{Tr}(B) $$

(c) The trace of $AB$ and $BA$ is the same.

$$ \text{Tr}(AB) = \text{Tr}(BA) $$

(c’) Cyclic Property: From the above fact, it can be known that the following equation holds.

$$ \text{Tr}(ABC) = \text{Tr}(BCA) = \text{Tr}(CAB) $$

(d) The trace of $A$ and $A^{T}$ is the same.

$$ \text{Tr}(A) = \text{Tr}(A^{T}) $$