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. In this thesis, we show that the commutator XY − Y X in the above inequality can be replaced by the product XY − Y XT for real matrices X and Y , where XT denotes the transpose of X. The proof is given in Chapter 2. We also give the characterization of those pairs of matrices that satisfy the inequality with equality in Chapter 3. Audenaert showed that for any n × n complex matrices X and Y , the above inequality can be strengthened as kXY − Y XkF ≤ √ 2kXkF kY k(2),2, where k·k(2),2 denotes the (2, 2)-norm. In Chapter 4 we show that the commutator XY −Y X in this inequality can also be replaced by the product XY − Y XT for real matrices X and Y . Those pairs of matrices which satisfy the inequality with equality are also characterized.