Given an matrix over , define , where bars denote the complex modulus. Prove the following for all and all .

Say and .

We have by the triangle inequality. Rearranging, we have as desired.

Now latex \sum_{i,j} \sum_k |a_{i,k}|b_{k,j}|$ by the triangle inequality. Now since all the new terms are positive, and rearranging, we have .

Finally, we have .