Analytic forms of eigenvalues and eigenvectors of 3x3 transition matrix

Dear sci.math.num-analysis

I hope this is a reasonable place for this post, apologies otherwise.

I am working on some Markov models to describe voltage-dependent
conformational changes in membrane proteins.

One of the branches is equivalent to a linear three compartment model
(C1<=>C2<=>C3), with fixed transition rates (eg a12,a23,a32, a21) for
any one voltage.

The transition matrix for the ODE's comes out to be

C1 | -a12 a21 0 |

C2 | a2 -(a23+a21) a32 |

C3 | 0 a23 -a32 |

the final form of equations is A1exp(-t/tau1) + A2exp(-t/tau2) +
A3exp(-t/tau3) + C
where tau 1-3 are the inverted eigenvalues of the matrix.

I can work out how to derive the eigenvalues analytically (and
numerically), but would very much like to know how to derive the
eigenvectors that determine the amplitudes of A1-3 - I have an old
solved solution, that I am a little uncertain of. My understanding is
that the amplitudes will be dependent on the initial conditions (ie
how much is in each compartment to start with), but I would like to be
able to quantify this analytically, to compare with experimental data.

I have access to Maxima.

With many thanks for your attention,

Chris French
Neurology
Royal Melbourne Hospital
.

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