Jordan block

A Jordan block is the smallest non-diagonalizable building piece. A Jordan block J_k(lambda) of size k is a k-by-k matrix with the eigenvalue lambda repeated all the way down the diagonal, 1s on the superdiagonal (the spots just to the upper-right of the diagonal), and zeros everywhere else. A size-1 block is just the scalar [lambda].

Each Jordan block has a single eigenvalue lambda whose algebraic multiplicity is k (it appears k times on the diagonal) but whose geometric multiplicity is exactly 1 — the eigenspace is one-dimensional, spanned by the first standard basis vector. That gap between algebraic and geometric multiplicity is precisely what diagonalization cannot fix, and what the off-diagonal 1s record.

Why indecomposable: a Jordan block cannot be split into a direct sum of smaller invariant pieces; it is the atom of the theory. The whole Jordan canonical form is just a direct sum (block-diagonal stack) of these atoms. Writing J_k(lambda) = lambda*I + N, where N is the nilpotent shift with 1s on the superdiagonal, separates the 'scaling' part lambda*I from the 'shifting' part N and makes formulas like powers and exp easy.

A caveat about convention: some authors put the 1s on the SUBdiagonal instead. The two conventions give transposed (hence similar) blocks, so the classification is the same; just check which your book uses before reading off entries.