Show that B=A - aXX^T is symmetric and rank B ≤ r-1

[JerryKelly]

If A is a symmetric nxn mx of rank<=r>=1 and X is a unit eigenvector of A, with eigenvalue a not= 0, let B=A - aXX^T. Show that B is symmetric and that N(A) is a proper subspace of N(B). Conclude that rank B=<r-1.
i could show X is in N(B) but not in N(A). Does anyone know how I can prove it in general? Then,I could prove N(A) is a proper subspace of N(B). Thanks!

 

[JerryKelly]

I figured it out already.