invariant direct-sum decomposition

Suppose you can break V into separate invariant pieces that fit together with no overlap and no gaps — V = W1 + W2 + ... + Wk as a direct sum, with each Wi invariant under T. Then T acts on each piece without mixing them: it is like running k independent smaller operators side by side.

Choose a basis by stacking bases of W1, then W2, and so on. In that basis the matrix of T is block-diagonal: square blocks down the diagonal, zeros everywhere else. The i-th block is exactly the matrix of the restriction T|Wi. The big problem of understanding T splits into k smaller, unrelated problems.

This is the whole goal of canonical forms. Diagonalization is the extreme case where every block is 1 by 1; Jordan form is the case where blocks are single Jordan blocks. Not every operator decomposes finely — but the primary decomposition theorem guarantees at least a coarse splitting along the prime-power factors of the minimal polynomial.