Matrix Exponential Fréchet Sensitivity

The matrix exponential, exp(A), generalizes the ordinary exponential function to matrices. A critical question is how the matrix exponential responds to a small perturbation in a particular direction E. We define the Fréchet derivative of the matrix exponential at A in the direction E as the linear map L such that:

Experiment

You can interact with the Fréchet derivative visually in the Matrix Exponential Fréchet Sensitivity module.

1. Set Matrix to "Jordan".
2. Set Direction to "Lower".
3. Set Time to 1 and Direction scale to 1.
4. Set Entry to "01".
5. Observe the "Fréchet derivative entry" which reads ~0.16666666666666666, confirming our derivation. Note the "Commuting shortcut entry" incorrectly predicts 0.0 because the matrices do not commute.