39. Trace

The trace of a square matrix $𝐴$ is the sum of its main diagonal entries:

39.1. Properties

• Linearity: $\text{tr}\left(𝐴+𝐵\right)=\text{tr}\left(𝐴\right)+\text{tr}\left(𝐵\right)$ and $\text{tr}\left(𝑐𝐴\right)=𝑐\text{tr}\left(𝐴\right)$
• Transpose: $\text{tr}\left({𝐴}^{𝑇}\right)=\text{tr}\left(𝐴\right)$

• Cyclic property: $\text{tr}\left(𝐴𝐵\right)=\text{tr}\left(𝐵𝐴\right)$

  • More generally: $\text{tr}\left(𝐴𝐵𝐶\right)=\text{tr}\left(𝐵𝐶𝐴\right)=\text{tr}\left(𝐶𝐴𝐵\right)$

• Similarity invariance: $\text{tr}\left({𝑃}^{-1}𝐴𝑃\right)=\text{tr}\left(𝐴\right)$

  • The trace doesn’t change under change of basis
• Sum of eigenvalues: $\text{tr}\left(𝐴\right)={\sum }_{𝑖=1}^{𝑛}{𝜆}_{𝑖}$ (counted with multiplicity, including complex)
• Identity: $\text{tr}\left({𝐼}_{𝑛}\right)=𝑛$

39.2. Trace as inner product

For matrices, the Frobenius inner product is defined via the trace:

It induces the Frobenius norm $\Vert 𝐴{\Vert }_{𝐹}=\sqrt{\text{tr}\left({𝐴}^{𝑇}𝐴\right)}$.

39.3. Where it shows up

• Sum of eigenvalues — quick check via $\text{tr}\left(𝐴\right)$
• Variance / covariance: $\text{tr}\left(\text{Cov}\right)$ = total variance across all dimensions
• Loss functions in machine learning — $\Vert 𝑦-𝑋𝛽{\Vert }^2=\text{tr}\left(\left(𝑦-𝑋𝛽\right){\left(𝑦-𝑋𝛽\right)}^{𝑇}\right)$
• Quantum mechanics — expectation values: $⟨\widehat{𝑂}⟩=\text{tr}\left(𝜌\widehat{𝑂}\right)$
• Characteristic polynomial: trace appears as a coefficient