Question 4.E.5.11: Sylvester’s law of nullity, given by James J. Sylvester in 1......
Sylvester’s law of nullity, given by James J. Sylvester in 1884, states that for square matrices A and B,
max \{ν(A), ν(B)\} ≤ ν(AB) ≤ ν(A) + ν(B),
where ν(\star) = dim N (\star) denotes the nullity.
(a) Establish the validity of Sylvester’s law.
(b) Show Sylvester’s law is not valid for rectangular matrices because ν(A) > ν(AB) is possible. Is ν(B) > ν(AB) possible?
(a) First notice that N (B) ⊆ N (AB) (Exercise 4.2.12) for all conformable A and B, so, by (4.4.5),
dim \mathcal{M} ≤ dim \mathcal{N}. (4.4.5)
dim N (B) ≤ dim N (AB), or ν(B) ≤ ν(AB), is always true—this also answers the second half of part (b). If A and B are both n × n, then the rank-plus-nullity theorem together with (4.5.2) produces
rank (AB) ≤ min {rank (A), rank (B)} , (4.5.2)
ν(A) = dim N (A) = n − rank (A) ≤ n − rank (AB) = dim N (AB) = ν(AB),
so, together with the first observation, we have max \{ν(A), ν(B)\} ≤ ν(AB). The rank-plus-nullity theorem applied to (4.5.3) yields ν(AB) ≤ ν(A) + ν(B).
rank (A) + rank (B) − n ≤ rank (AB). (4.5.3)
(b) To see that ν(A) > ν(AB) is possible for rectangular matrices, consider A = \begin{pmatrix}1&1\end{pmatrix} and B = \begin{pmatrix}1\\1\end{pmatrix}.