A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue

Question

When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The aesthetic here being that mathematicians look at fixed points to exploit symmetry to find solutions.

If one considers a 2x2 first order matrix linear differential equation of the form $$X'=AX$$ when $A$ has a repeated eigenvalue with only a one-dimensional eigenspace, one can still discover what needs doing using the approximate fixed point idea while appealing to an analogy with the one-dimensional procedure of "reduction of order". This is done here, for example, in the way most of us probably teach it.

I always feel like I'm making a stretch when teaching this particular material, and was wondering if anyone had a more global explanation for the appearance of generalized eigenvectors here? I always felt that there should be, but can't think of a more natural way to explain this than the process found in the link above.

Answer 1

So, there is some point where they just propose multiplying the given solution by $t$ and then work out the generalized e-vector of order two condition. Pretty in its way, but I prefer the perspective of the matrix exponential.

Define $e^{tA}=I+tA+(t^2/2)A^2+ \cdots$. It is easy to show that the matrix exponential is a solution matrix and it is an invertible matrix. Thus, the columns of the matrix exponential form a fundamental solution set; contained in $e^{tA}$ are all the solutions of $x'=Ax$.

The question then, how to actually calculate the columns without explicit computation of the series. One technique is suggested by the following formula: $$ e^{tA} = e^{\lambda t}(I+t(A-\lambda I)+(t^2/2)(A-\lambda I)^2+(t^3/3!)(A-\lambda I)^3+ \cdots ) $$ This identity holds for any complex scalar $\lambda$. However, it is most useful when we consider eigenvectors and then generalized e-vectors. It produces the usual cases given in the document linked by the OP as well as an infinite family of possibilities for higher order problems. Of course, the Jordan cannonical form completes the thought of this discussion. I can say more, but this is standard stuff.

All of this said, I'm not sure this is the motivation you seek. Another idea, $y''=0$ has solution $y = c_1+c_2t$ by calculus I. Then do reduction of order and study the structure of the resulting solutions. You'll find our beloved eigenvector solutions as well as generalized e-vectors of order 2. This might motivate. You might look at: https://math.stackexchange.com/q/206967/36530 as the inquiry and discussion there is very much related to your question.