Let be the Jordan matrix
Determine the matrix for some differentiable function .
If we experiment by exponentiating the matrix , we will discover two properties:
and the process continues until
2) for .
Hence is called the shift matrix (you shift the 1-diagonal, building up a trail of zeroes the more you exponentiate). Property 2 illustrates the intrinsic nilpotent property of shift matrices.
The exponent of is computed as follows:
(essentially the binomial expansion).
To illustrate the mechanics:
We expand the matrix function
for some point where the derivatives of (with respect to ) exist.
We substitute the exponents of (giving it the shifting property) to the expanded matrix function , which is actually a sum of matrices.
Collapsing this sum will yield the matrix
Check out my other notes at Proof, Disproof, and Derivation