Trace

矩阵的迹 / Matrix Trace

定义 / Definition

迹（trace）: 一个方阵 $\mathbf{A}$ 的迹是其对角线元素之和，用 $\mathrm{tr}(\mathbf{A})$ 表示 Trace: The trace of a square matrix $\mathbf{A}$ is the sum of its diagonal elements, denoted as $\mathrm{tr}(\mathbf{A})$

$$\mathrm{tr}(\mathbf{A})=\sum _{i=1}^n{a}_{ii}$$

其中，$a_{ii}$ 表示矩阵 $\mathbf{A}$ 的第 $i$ 个对角线元素

where $a_{ii}$ represents the $i$-th diagonal element of matrix $\mathbf{A}$

迹的性质 / Properties of Trace

1. 线性性 / Linearity

若 $\mathbf{A}$ 和 $\mathbf{B}$ 是两个 $n\times n$ 矩阵，且 $c$ 是一个标量，则

If $\mathbf{A}$ and $\mathbf{B}$ are two $n\times n$ matrices, and $c$ is a scalar, then

$$\mathrm{tr}(\mathbf{A}+\mathbf{B})=\mathrm{tr}(\mathbf{A})+\mathrm{tr}(\mathbf{B})$$ $$\mathrm{tr}(c\mathbf{A})=c\cdot \mathrm{tr}(\mathbf{A})$$

2. 迹的相似不变性 / Invariance under Similarity

如果 $\mathbf{A}$ 和 $\mathbf{B}$ 是相似矩阵，即存在可逆矩阵 $\mathbf{P}$ 使得 $\mathbf{B}={\mathbf{P}}^{-1}\mathbf{AP}$，则

If $\mathbf{A}$ and $\mathbf{B}$ are similar matrices, i.e., there exists an invertible matrix $\mathbf{P}$ such that $\mathbf{B}={\mathbf{P}}^{-1}\mathbf{AP}$, then

$$\mathrm{tr}(\mathbf{A})=\mathrm{tr}(\mathbf{B})$$

3. 迹与转置矩阵 / Trace and Transpose Matrix

对于任意矩阵 $\mathbf{A}$，其转置矩阵 ${\mathbf{A}}^{\mathrm{T}}$ 的迹与其本身相等，即

For any matrix $\mathbf{A}$, the trace of its transpose ${\mathbf{A}}^{\mathrm{T}}$ is equal to its own trace, i.e.,

$$\mathrm{tr}(\mathbf{A})=\mathrm{tr}({\mathbf{A}}^{\mathrm{T}})$$

4. 迹与矩阵乘积 / Trace and Matrix Product

对于任意 $n\times n$ 矩阵 $\mathbf{A}$ 和 $\mathbf{B}$，有

For any $n\times n$ matrices $\mathbf{A}$ and $\mathbf{B}$, we have

$$\mathrm{tr}(\mathbf{AB})=\mathrm{tr}(\mathbf{BA})$$

这个性质可以推广到多个矩阵的乘积，即

This property can be generalized to the product of multiple matrices, i.e.,

$$\mathrm{tr}(\mathbf{ABC})=\mathrm{tr}(\mathbf{CAB})=\mathrm{tr}(\mathbf{BCA})$$

5. 迹与特征值 / Trace and Eigenvalues

对于一个 $n\times n$ 矩阵 $\mathbf{A}$，其特征值 ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ 的和等于其迹

For an $n\times n$ matrix $\mathbf{A}$, the sum of its eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ equals its trace

$$\mathrm{tr}(\mathbf{A})=\sum _{i=1}^n{\lambda }_i$$

6. 迹的迹（trace of the trace）/ Trace of the Trace

若 $\mathbf{A}$ 是一个 $n\times n$ 方阵，则

If $\mathbf{A}$ is an $n\times n$ square matrix, then

$$\mathrm{tr}(\mathrm{tr}(\mathbf{A}))=\mathrm{tr}(\mathbf{A})$$

证明 / Proofs

迹的相似不变性 / Invariance under Similarity

如果 $\mathbf{A}$ 和 $\mathbf{B}$ 是相似矩阵，即存在可逆矩阵 $\mathbf{P}$ 使得 $\mathbf{B}={\mathbf{P}}^{-1}\mathbf{AP}$，则

If $\mathbf{A}$ and $\mathbf{B}$ are similar matrices, i.e., there exists an invertible matrix $\mathbf{P}$ such that $\mathbf{B}={\mathbf{P}}^{-1}\mathbf{AP}$, then

计算 $\mathbf{B}$ 的迹

Compute the trace of $\mathbf{B}$

$$\mathrm{tr}(\mathbf{B})=\mathrm{tr}({\mathbf{P}}^{-1}\mathbf{AP})$$

利用迹的性质 $\mathrm{tr}(\mathbf{XY})=\mathrm{tr}(\mathbf{YX})$

Using the property of trace $\mathrm{tr}(\mathbf{XY})=\mathrm{tr}(\mathbf{YX})$

$$\mathrm{tr}({\mathbf{P}}^{-1}\mathbf{AP})=\mathrm{tr}(\mathbf{AP}{\mathbf{P}}^{-1})=\mathrm{tr}(\mathbf{A})$$

因此，相似矩阵的迹相等

Therefore, the trace of similar matrices is equal

迹与特征值的和 / Trace and Sum of Eigenvalues

设 $\mathbf{A}$ 是一个 $n\times n$ 方阵，${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ 是 $\mathbf{A}$ 的特征值

Let $\mathbf{A}$ be an $n\times n$ square matrix, and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be the eigenvalues of $\mathbf{A}$

根据特征多项式 $\det (\mathbf{A}-\lambda \mathbf{I})=0$，我们知道 $\mathbf{A}$ 的特征值是 $\mathbf{A}$ 的特征多项式的根

According to the characteristic polynomial $\det (\mathbf{A}-\lambda \mathbf{I})=0$, we know that the eigenvalues of $\mathbf{A}$ are the roots of the characteristic polynomial of $\mathbf{A}$

特征多项式可以表示为

The characteristic polynomial can be written as

$$\det (\mathbf{A}-\lambda \mathbf{I})=(-1{)}^n({\lambda }^n-(\mathrm{tr}(\mathbf{A})){\lambda }^{n-1}+\cdots +\det (\mathbf{A}))$$

其中，$\mathrm{tr}(\mathbf{A})$ 是 $\mathbf{A}$ 的迹，表示 $\mathbf{A}$ 的特征值之和

where $\mathrm{tr}(\mathbf{A})$ is the trace of $\mathbf{A}$, representing the sum of the eigenvalues of $\mathbf{A}$

结论 / Conclusion

$$\mathrm{tr}(\mathbf{A})=\sum _{i=1}^n{a}_{ii}=\sum _{i=1}^n{\lambda }_i$$

迹是矩阵的重要性质之一，等于对角线元素的和，也等于特征值的和

Trace is one of the important properties of a matrix, equal to the sum of the diagonal elements and the sum of the eigenvalues

Derivatives Involving Trace / 迹相关的导数

1. $\frac{\partial }{\partial \mathbf{A}}\mathrm{tr}(\mathbf{AX})={\mathbf{X}}^T$

   When differentiating the trace of the product of matrices $\mathbf{A}$ and $\mathbf{X}$, with respect to $\mathbf{A}$, the result is the transpose of $\mathbf{X}$.

   对 $\mathrm{tr}(\mathbf{AX})$ 关于 $\mathbf{A}$ 求导时，结果为 $\mathbf{X}$ 的转置。

2. $\frac{\partial }{\partial \mathbf{A}}\mathrm{tr}(\mathbf{X}{\mathbf{A}}^T)=\mathbf{X}$

   Differentiating the trace of the product of $\mathbf{X}$ and ${\mathbf{A}}^T$ with respect to $\mathbf{A}$ results in $\mathbf{X}$.

   对 $\mathrm{tr}(\mathbf{X}{\mathbf{A}}^T)$ 关于 $\mathbf{A}$ 求导时，结果为 $\mathbf{X}$。

3. $\frac{\partial }{\partial \mathbf{A}}\mathrm{tr}(\mathbf{AX}{\mathbf{A}}^T)=\mathbf{A}(\mathbf{X}+{\mathbf{X}}^T)$

   When differentiating the trace of $\mathbf{AX}{\mathbf{A}}^T$ with respect to $\mathbf{A}$, the result is $\mathbf{A}$ multiplied by the sum of $\mathbf{X}$ and its transpose.

   对 $\mathrm{tr}(\mathbf{AX}{\mathbf{A}}^T)$ 关于 $\mathbf{A}$ 求导时，结果为 $\mathbf{A}$ 乘以 $\mathbf{X}$ 和其转置之和。

Additional Notes / 额外说明

• The trace of a matrix is invariant under cyclic permutations, which can often simplify expressions and calculations. For example, in certain optimization problems, the cyclic property can be used to manipulate the objective function.
• 矩阵的迹在循环置换下不变，这通常可以简化表达式和计算。例如，在某些优化问题中，循环性质可用于操作目标函数。

This overview covers the basic concepts and properties of matrix trace and its derivatives, which are essential tools in various fields such as machine learning, statistics, and applied mathematics. 这份概述涵盖了矩阵迹及其导数的基本概念和性质，这些是机器学习、统计学和应用数学等领域的重要工具。