A linear transformation S: U → U is called skew-adjoint if S* = -S.
(a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on Rn.
(b) Under what conditions is S[x] = Ax skew-adjoint with respect to the inner product (x, y) = xT M y on Rn?
(c) Let L: U → U have an adjoint L*. Prove that L - L* is skew-adjoint.
(d) Explain why every linear operator L: U → U that has an adjoint L* can be written as the sum of a self-adjoint and a skew-adjoint operator.