Trace of Matrix

Before deriving the form of an optimal baseline in AC A2C, let’s understand some prerequisite knowledge: the trace of a matrix.

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In linear algebra, the trace is an important property of a square matrix, defined as the sum of elements on the main diagonal. Here’s a detailed explanation:

Definition

For an $n\times n$ square matrix $A=[a_{ij}]$, its trace is denoted as $\text{tr}(A)$, calculated by the formula:

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

Examples:

$$\begin{array}{rl}A& =\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=1+4=5\\ B& =\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]=1+5+9=15\end{array}$$

Properties

1. Linearity:

   $$\text{tr}(A+B)=\text{tr}(A)+\text{tr}(B),\text{tr}(cA)=c\cdot \text{tr}(A)$$

   where $A,B$ are square matrices of the same order, and $c$ is a scalar.

2. Commutativity of Matrix Multiplication:

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

   This holds true even when $AB$ and $BA$ have different dimensions (e.g., when $A$ is $m\times n$ and $B$ is $n\times m$).

3. Similarity Invariance: If $B=P^{-1}AP$ (similarity transformation), then $\text{tr}(B)=\text{tr}(A)$. This indicates that trace is a similarity invariant of matrices.

4. Relationship with Eigenvalues: The trace equals the sum of all eigenvalues (counting algebraic multiplicity). If the eigenvalues of $A$ are ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$, then:

   $$\text{tr}(A)={\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n$$

5. Other Properties:

   • Transpose doesn’t change the trace: $\text{tr}(A)=\text{tr}(A^{\top })$.
   • The trace of an identity matrix equals its order: $\text{tr}(I_n)=n$.

Applications

1. Eigenvalue Analysis: The trace and determinant together reflect the characteristics of a matrix (trace is the sum of eigenvalues, determinant is the product of eigenvalues).
2. Optimization Problems: In statistics and machine learning, the trace is used to measure the total variance of a covariance matrix.
3. Quantum Mechanics: The trace of a density matrix represents the total probability of a system (which must remain 1).

Example Verification

Commutativity Verification:

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, $B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, then:

$$\begin{array}{rl}AB& =\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right],\text{tr}(AB)=19+50=69\\ BA& =\left[\begin{array}{cc}23& 34\\ 31& 46\end{array}\right],\text{tr}(BA)=23+46=69\end{array}$$

The results are consistent, verifying that $\text{tr}(AB)=\text{tr}(BA)$.

Summary

The trace is the sum of the elements on the main diagonal of a square matrix. It has properties of linearity, similarity invariance, and is closely related to eigenvalues. It has wide applications in matrix analysis, optimization, and physics.