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# Double inequality involving matrix

Prove that

Solution

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.

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