How is the instantaneous rate of an instantaneous rate matrix determined?

In molecular evolution, I know that the instantaneous rate of the rate matrix is the limit of the rate as time approaches 0, but what I don't understand is how this rate is established or what I can do with this number.

For instance, if a discrete character is thought to have changed from 0 to 1 five times on a tree with a root age of 40 million years, is the proportion 5 / 40 million the same as the instantaneous rate of change? If not, how exactly are they different?

From a different perspective, let's assume an (absolute) instantaneous rate of 0.0001 and let's say I have a tree with a root age of 2 million years. Can I multiply 0.0001 by two million to get the expected number of transitions of that character across the tree?

• Mar 15 '21 at 19:28
• @MaximilianPress I saw this post. Although both questions concern rate matrices, they are nevertheless quite distinct. Mar 16 '21 at 20:10
• I agree; note that I didn't mark it as duplicate. However, they have a substantial overlap and it is worth annotating as such, for anyone coming across this question. Mar 16 '21 at 20:14