block-triangular form

Block-triangular form is the matrix shape forced the instant you have a single chain of nested invariant subspaces, even a partial one. Instead of scalars on and above the diagonal, you get whole square blocks down the diagonal and arbitrary blocks above, but all zeros in the blocks below.

Concretely, suppose {0} inside W1 inside W2 inside ... inside Wk = V is a chain of invariant subspaces, and you pick a basis adapted to it. Then the matrix of T is block-upper-triangular: the i-th diagonal block is exactly the matrix of the induced operator on the quotient Wi / W(i-1), and everything strictly below the block diagonal is zero. The off-diagonal blocks above record how T mixes one quotient into the lower ones.

This is strictly weaker than block-diagonal and that gap is the whole story of canonical forms. Block-triangular needs only invariance (a nested chain); block-diagonal additionally needs each piece to have an invariant complement (a true direct sum). Two clean consequences: the characteristic polynomial of T is the product of the characteristic polynomials of the diagonal blocks, and the determinant is the product of the diagonal-block determinants — the upper blocks never contribute.