  If $c_1(Z) \geq ... \geq c_n(Z)$ denote the Euclidean lengths of the column
vectors of any $n \times n$ matrix $Z,$ then a fundamental inequality related
to Hadamard products states that
  $$ \sum_{i=1}^k \sigma_i(X^*Y \circ B) \leq \sum_{i=1}^k c_i(X) c_i(Y)
\sigma_i(B) \qquad 1 \leq k \leq n,$$ where $\sigma_i(\cdot)$ is the $i$th
singular value. In this paper, we shall offer a simple proof of this result via
convexity arguments. In addition, this technique is applied to obtain some
further singular value inequalities as well.
