I understand how to solve homogeneous linear ODE systems with defective eigenvalues without utilizing [MATH]e^{At}[/MATH] by plugging an eigenvector (a.k.a., rank 1 generalized eigenvector) [MATH]v_{r1}[/MATH] into [MATH](A - \lambda I)v_{r2} = v_{r1}[/MATH], then [MATH](A - \lambda I)v_{r3} = v_{r2}[/MATH], and repeating the pattern recursively until all generalized eigenvectors are revealed one rank at a time (peeling the onion back layer by layer, to make a TOR analogy). The solution to the ODE is then formed by pairing the generalized eigenvectors of specific ranks with specific functions of [MATH]t[/MATH].
Where I get a bit confused is finding all of the same information via the [MATH]e^{At}[/MATH] procedure. Here is my book's explanation:
I have gathered that there's a typo in 3., where [MATH]v_j[/MATH] should only appear once and the other instances should be replaced with other generalized eigenvectors. I can sort that part out on my own. My confusion is over how to pair the generalized eigenvectors with the correct functions of t. The book's basis obtained in 2. is not sorted by rank in the way my [MATH]v_{r1}[/MATH], [MATH]v_{r2}[/MATH], and [MATH]v_{r3}[/MATH] are; even though the book does subscript with numbers, the order of vectors in a basis is arbitrary, while their placement in the solution to the ODE is not. Since this procedure obtains all generalized eigenvectors at once instead of peeling them off layer by layer, how do we know which function of t to associate with each?
Where I get a bit confused is finding all of the same information via the [MATH]e^{At}[/MATH] procedure. Here is my book's explanation:
I have gathered that there's a typo in 3., where [MATH]v_j[/MATH] should only appear once and the other instances should be replaced with other generalized eigenvectors. I can sort that part out on my own. My confusion is over how to pair the generalized eigenvectors with the correct functions of t. The book's basis obtained in 2. is not sorted by rank in the way my [MATH]v_{r1}[/MATH], [MATH]v_{r2}[/MATH], and [MATH]v_{r3}[/MATH] are; even though the book does subscript with numbers, the order of vectors in a basis is arbitrary, while their placement in the solution to the ODE is not. Since this procedure obtains all generalized eigenvectors at once instead of peeling them off layer by layer, how do we know which function of t to associate with each?