Nullspace of Matrix times projection matrix

Question

I'm given a matrix X, a matrix A and a projection matrix $ P_{N(A)} $ . What is the nullspace of $X \cdot P_{N(A)}$ ? My script says its $ N(X) \cap N(A) $. Is it true ? Why ? Thanks

Answer 1

First of all, any vector in $N(A)^\perp = R(A^\top)$ is in the nullspace of this matrix, since it projects to $0\in N(A)$. Next, any vector in $N(A)\cap N(X)$ is in the nullspace: If $v\in N(A)\cap N(X)$, then $P_{N(A)}v = v$, and so $X\cdot P_{N(A)}v = Xv = 0$. Thus, any linear combination of such vectors is in the nullspace, so we claim the nullspace is precisely $$N(A)\cap N(X) + R(A^\top).$$

We've argued that any vector in that subspace is in the nullspace. Why is that everything? Take any $v\in N(X\cdot P_{N(A)})$ and write $v=v_1+v_2$, where $v_1\in N(A)$ and $v_2\in N(A)^\perp = R(A^\top)$. Then $$0=(X\cdot P_{N(A)})v = (X\cdot P_{N(A)})v_1 + (X\cdot P_{N(A)})v_2 = Xv_1,$$ so $v_1\in N(X)\cap N(A)$, as desired.