Fix . Apply the matrix on thus:
Since is exactly the – th column of the previous equality can be rerwritten as
Since this holds for all we get and the left inequality follows.
For a random unit vector the coordinate of the vector is . It follows from Cauchy – Schwartz that
Summing over all ‘s till we find we conclude that, for every unit vector , it holds that is less than the right hand side. Taking supremum with respect to all the right hand side inequality follows.