*instantaneous*rate.

One of my sources is Ziheng Yang's

*Computational Molecular Evolution*. According to him (and wikipedia), this model can be represented by a

*substitution-rate matrix*which is usually called Q:

Yang uses λ in place of the often-used α. For this post we'll follow along with him.

As I posted about the other day, we need to have equations that will give us the probabilities for any time t. These should be recognizable:

Yang calls these

`p`_{0}(t)

and `p`_{1}(t)

.The corresponding transition-probability matrix is designated P(t) and it is:

And if we had P for one unit of evolutionary time, then we could do matrix multiplication (see here) to generate P for any time.

Now, Yang (and other sources) also says that:

And I've been trying to wrap my head around

*that*, i.e. matrix exponentiation. Luckily, I've been getting into Gilbert Strang's MIT lectures on linear algebra (here). So I can appreciate what the

*inverse*of a matrix is, such that:

where I is just the identity matrix of the same size and shape as Q. Now, suppose we can

*diagonalize*Q. That is, we can find an invertible matrix U and a diagonal matrix Λ such that:

A diagonal matrix means that the matrix has non-zero values only on the diagonal, e.g.:

And that means that

`Λ Λ`

is simply:Look what happens when we do something like:

We only have to deal with Λ, and it's straightforward. Now for the real magic. I am reliably informed that this trick works for any function. For example:

where we have:

And, most impressive of all, U and U

^{-1}are the right and left eigenvectors of Q, and the coefficients of Λ are its eigenvalues! We've looked at that in detail previously(here).

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