I understand how to go about finding the eigenvalues and eigenvectors in principle, in order to distinguish the phugoid modes, but the first exercise in Lavretsky & Wise tells to compute those values analytically. All examples online first pick numerical values and then compute the eigenvalue, as numerical values. Am I misunderstanding the problem in Lavretsky & Wise, or are they telling to express the eigenvalues algebraically in terms of the entries of the general matrix of longitudinal dynamics? And if they are, are there some observations that simplify the task, or must one plough through the roots of a quartic polynomial brute force?
are they telling to express the eigenvalues algebraically in terms of the entries of the general matrix of longitudinal dynamics?
The way I read the problem yes they are. Although I'm not familiar with this book, what they are probably trying to get you to see is some relationships between the variables. E.g. if you solve this numerically, you've got to pick values for all of the variables. You have to pick an airspeed V0, turn the crank, and an eigenvalue will pop out. Now ask yourself, how does increasing airspeed (or some other variable) affect the eigevalues? Well you have no idea. You have pick a whole bunch of other values of V0, compute a whole bunch more eigenvalues, make some plots, and then try to discern if its is a linear relationship, quadratic, etc. But if you solve it analytically, the answer will just pop out. V0 will be right there in the answer, and you can see exactly how it affects each mode.
And if they are, are there some observations that simplify the task,
Well you've got a bunch of zeros in the bottom row. That could be of great help in expanding the determinate.