trace

Topics

• matrices
• linear algebra

Let $A\in {\mathbb{R}}^{n\times n}$ (square matrix), the trace of $A$ (sum of diagonal elements) is defined as,

$$Tr(A)=\sum _{i=1}^n{A}_{ii}$$

Properties

• Linear mapping
• Invariant under cyclic permutations i.e. $Tr(ABC)=Tr(CAB)=Tr(BCA)$
• Invariant under transpose i.e. $Tr(A^{T}B)=Tr(B^{T}A)=Tr(A{B}^T)$
• For a symmetric matrix $A$, we have $Tr(A)={\sum }_{i=1}^n{\lambda }_i$ (sum of eigenvalues)
  • For a symmetric matrix, $Tr(A)=Tr(QD{Q}^T)$ by spectral decomposition
  • Using the cyclic permutation property of trace, we have: $Tr(A)=Tr(QD{Q}^T)=Tr(Q{Q}^{T}D)=Tr(D)={\sum }_{i=1}^n{\lambda }_i$

from numpy import trace
 
A = np.array([[1,2,3],
              [4,5,6],
              [7,8,9]])
t = trace(A)  # 1 + 5 + 9 = 15

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