indecomposable operator

An operator is indecomposable if you cannot break its space into two nontrivial invariant pieces — there is no way to write V = W1 (+) W2 with both Wi invariant, nonzero, and proper. The operator is one solid lump; it refuses to split into independent parallel actions.

Be careful to separate two notions. Indecomposable means it has no invariant direct-sum splitting. Irreducible (or simple) means it has no nontrivial invariant subspace at all. Irreducible implies indecomposable, but not the reverse: a single Jordan block has plenty of invariant subspaces (the chain of partial spans) yet none of them has an invariant complement, so it cannot be decomposed.

The single Jordan block J_lambda is the model indecomposable operator over an algebraically closed field. The deep payoff is uniqueness: the Krull-Schmidt theorem says any finite-dimensional operator decomposes into indecomposables in an essentially unique way, blocks and multiplicities determined up to reordering and isomorphism. That uniqueness is precisely why Jordan canonical form is a genuine invariant of the operator.