39. Trace

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

39.1. Properties

• Linearity: and
• Transpose:

• Cyclic property:

  • More generally:

• Similarity invariance:

  • The trace doesn’t change under change of basis
• Sum of eigenvalues: (counted with multiplicity, including complex)
• Identity:

39.2. Trace as inner product

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

It induces the Frobenius norm .

39.3. Where it shows up

• Sum of eigenvalues — quick check via
• Variance / covariance: = total variance across all dimensions
• Loss functions in machine learning —
• Quantum mechanics — expectation values:
• Characteristic polynomial: trace appears as a coefficient