It has been conjectured and proved that kXY − Y XkF ≤ √ 2kXkF kY kF for any n × n complex matrices X and Y , where k · kF denotes the Frobenius norm. A characterization of those pairs of matrices that satisfy the inequality with equality has also been found. Thereafter, Audenaert gave another proof for the inequality by means of what he called the matrix version of variance. Based on his proof, we find another proof for the equality cases in Chapter 2. Audenaert also showed that kXY − Y XkF ≤ √ 2kXkF kY k(2),2, where k · k(2),2 denotes the (2, 2)-norm. In Chapter 3 we characterize the pairs of matrices which satisfy the inequality with equality. Furthermore, we extend this inequality to other Schatten p-norms in Chapter 4. On the other hand, B¨ottcher and Wenzel proved that for any unitarily invariant norm k · k, sup  kXY − Y Xk kXkkY k : X and Y are n × n nonzero complex matrices  = C ≥ √ 2. They also asked whether the Frobenius norm is the only one having such property. In Chapter 5 we answer the question by showing that the dual norm of the (2, 2)- norm also has the property that C = √ 2.