Let T: V Vbe a linear operator on an inner product space V of finite dimension.

Question:

Let T: V → Vbe a linear operator on an inner product space V of finite dimension. Show that the following are equivalent.
(1) (v, T(w)) = -(T(v), w) for all v and w in V.
(2) MB(T) is skew-symmetric for every orthonormal basis B.
(3) MB(T) is skew-symmetric for some orthonormal basis B.
Such operators T are called skew-symmetfic operators.