Variance bounds, with an application to norm bounds for commutators

Murthy and Sethi (Sankhya Ser B \textbf{27}, 201--210 (1965)) gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and, more importantly, to the matrix case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range, and introduce the new concept of radius of a matrix. We also give a new and simplified proof for a sharp upper bound on the Frobenius norm of commutators recently proven by B\"ottcher and Wenzel (Lin.\ Alg. Appl. \textbf{429} (2008) 1864--1885) and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced. As an immediate application of our variance bounds we obtain stronger versions of B\"ottcher and Wenzel's upper bound.