Editing Trace (linear algebra) (section)

=== Derivative relationships ===
If {{math|'''ΔA'''}} is a square matrix with small entries and {{math|'''I'''}} denotes the [[identity matrix]], then we have approximately

<math display="block">\det(\mathbf{I}+\mathbf{\Delta A})\approx 1 + \operatorname{tr}(\mathbf{\Delta A}).</math>

Precisely this means that the trace is the [[derivative]] of the [[determinant]] function at the identity matrix. [[Jacobi's formula]] 

<math display="block">d\det(\mathbf{A}) = \operatorname{tr} \big(\operatorname{adj}(\mathbf{A})\cdot d\mathbf{A}\big)</math>

is more general and describes the [[Differential (infinitesimal)|differential]] of the determinant at an arbitrary square matrix, in terms of the trace and the [[Adjugate matrix|adjugate]] of the matrix.

From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the [[matrix exponential]] function, and the determinant:<math display="block">\det(\exp(\mathbf{A})) = \exp(\operatorname{tr}(\mathbf{A})).</math>

A related characterization of the trace applies to linear [[vector field]]s. Given a matrix {{math|'''A'''}}, define a vector field {{math|'''F'''}} on {{math|'''R'''<sup>''n''</sup>}} by {{math|1='''F'''('''x''') = '''Ax'''}}. The components of this vector field are linear functions (given by the rows of {{math|'''A'''}}). Its [[divergence]] {{math|div '''F'''}} is a constant function, whose value is equal to {{math|tr('''A''')}}.

By the [[divergence theorem]], one can interpret this in terms of flows: if {{math|'''F'''('''x''')}} represents the velocity of a fluid at location {{math|'''x'''}} and {{mvar|U}} is a region in {{math|'''R'''<sup>''n''</sup>}}, the [[flow network|net flow]] of the fluid out of {{mvar|U}} is given by {{math|tr('''A''') · vol(''U'')}}, where {{math|vol(''U'')}} is the [[volume]] of {{mvar|U}}.

The trace is a linear operator, hence it commutes with the derivative:
<math display="block">d \operatorname{tr} (\mathbf{X}) = \operatorname{tr}(d\mathbf{X}) .</math>