대각합

Definition

Let $n\times n$ be a matrix 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 considered a function. Let $M_{n\times n}(\mathbb{R})$ be the set of $n\times n$ matrices with real number components. Then $\text{Tr}$ is a function defined as follows.

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

If the trace is a function, there’s nothing to stop us from differentiating it.

Properties

Let $A = [a_{ij}]$ , $B=[b_{ij}]$ , $C$ be $n \times n$ matrices, and let $k$ be a constant.

(a) The trace of a scalar multiple and the scalar multiple of a trace are equal.

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

(b) The trace of a sum and the sum of traces are equal.

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

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

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

(c) The traces of $AB$ and $BA$ are equal.

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

(c’) Cyclic Property: From this fact, we can see that the following equation holds.

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

(c") In fact, this also holds for any $A \in \mathbb{R}^{n \times k}$ and $B \in \mathbb{R}^{k \times n}$ .

(d) The traces of $A$ and its transpose matrix $A^{\mathsf{T}}$ are equal.

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

(e) The following holds.

$\sum\limits_{i,k} a_{ik}b_{ki} = \text{Tr}(AB) = \text{Tr}(B^{\mathsf{T}} A^{\mathsf{T}})$

$\sum\limits_{i,k} a_{ik}b_{ik} = \text{Tr}(AB^{\mathsf{T}}) = \text{Tr}(A^{\mathsf{T}} B)$

Proof

(e)

The $(i,j)$ component of the product of matrices $A$ , $B$ can be expressed as follows.

$[AB]_{ij} = \sum\limits_{k} a_{ik}b_{kj}$

Since the trace is the sum of components where $i=j$ ,

$\Tr(AB) = \sum\limits_{i,k} a_{ik}b_{ki}$

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