An $n \times n$ skew-symmetric matrix $A$ is converted into an upper-Hessenberg form $B$, say, by Householder reflections.

(a) Describe each step of the procedure and observe that $B$ is tridiagonal. Your algorithm should take advantage of the special form of $A$ to reduce the number of computations.

(b) Compare the cost (counting only products and looking only at the leading term) of converting a skew-symmetric and a symmetric matrix to an upper-Hessenberg form using Householder reflections.