Trace

Definition of trace:

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

• Property:
  • $\text{tr}(AB)=\text{tr}(BA)$
  • $\text{tr}(A)={\sum }_{i=1}^n{\lambda }_i$
• Interpretation:
  • Consider a linear vector field defined $\vec{F}=A\vec{X}$. Then, trace = divergence of a vector field.
  • Zero trace = incompressible. Positive trace: expanding fluid. Negative trace: contracting fluid.

Eigenvalues