Q:

Units of eigenvalues

Suppose you have the system $\bf x' = \bf Ax$, where $\bf x$ is a vector and $\bf A$ is a matrix. What are the units of the eigenvalues of $\bf A$? I think they should be $1/t$ but I'm not sure how to verify this. Can you give me a starting point?

A:

Your intuition is correct, they should be 1/(the dimension of what you are taking the derivative with respect to).  If your derivative is with respect to time, then it should be $1/t$.  If your derivative is with respect to something unitless, the eigenvalue will be unitless, too.  Write it as $\frac {dx}{dt}=Ax$  The units of $A$ are then $1/t$ and so the eigenvalues will be, too.  If you are taking derivatives with respect to time, maybe the question belongs at physics.

