The above argument can be slightly restated as follows: let "λ" be an eigenvalue of "A", corresponding to some eigenspace "Vλ". "A" induces an operator "T" on the quotient space C"n"/"Vλ". This operator is precisely the "A"22 submatrix from above. As before, "T" would have an eigenspace, say "Wμ" ⊂ C"n" modulo "Vλ". Notice the preimage of "Wμ" under the quotient map is an invariant subspace of "A" that contains "Vλ". Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that "A" stabilizes.