Question: The members of the basis associated with the Jordan Canonical Form are often referred to as occurring in chains-this terminology arising from the following interpretation. If the geometric multiplicity of an eigenvalue λ is m, then m linearly independent eigenvectors are part of the basis. Each of these eigenvectors satisfies [A-λI]e = 0 and each is considered as the first element of one of m separate chains associated with the eigenvalue λ. The next member of the chain associated with e is a vector f such that[A-λI]f = e The chain continues with a g satisfying [A-λI]g = f, and so forth until the chain ends. The original m eigenvectors generate m separate chains, which may have different lengths.
Given a matrix J in Jordan form with m blocks associated with the eigenvalue λ, find the m eigenvectors of J. Also find the vectors in the chain associated with each eigenvector.