induced map on a quotient

Suppose an operator T : V -> V preserves a subspace U, meaning U is T-invariant. Then T does not just restrict to U from below — it also descends to the quotient space V/U from above. The induced map, written T-bar, acts on cosets by the obvious rule: T-bar(v + U) = T(v) + U. It is the way T looks once you ignore everything happening inside U.

The reason this is allowed is exactly invariance. For the rule to be well defined, two representatives of the same coset must give the same answer. If v and v' differ by an element u of U, then T(v) and T(v') differ by T(u), which lies in U precisely because U is invariant — so they represent the same coset of V/U. Without invariance the formula would be ambiguous; with it, T-bar is a genuine, well-defined operator on V/U.

Restriction and induced map are the two halves of one operator. T|_U captures what T does inside U; T-bar captures what T does to the rest, modulo U. In matrix terms, with a basis of U extended to V, the block form [A, B; 0, C] displays both at once: A is the restriction to U, and C is the matrix of the induced map on V/U. The block B is the only information not seen by either piece.

This is the standard tool for peeling an operator apart inductively. Find one invariant subspace U, understand T|_U and the smaller operator T-bar on V/U, and recurse. The eigenvalues of T are the eigenvalues of T|_U together with those of T-bar, which is how triangularization and the Jordan form get built one invariant layer at a time.