1. Useful versions of the Ahlswede-Winter Inequality
Theorem 1 Let be a random, symmetric, positive semi-definite matrix such that . Suppose for some fixed scalar . Let be independent copies of (i.e., independently sampled matrices with the same distribution as ). For any , we have
This event is equivalent to the sample average having minimum eigenvalue at least and maximum eigenvalue at most .
Proof: We apply the Ahlswede-Winter inequality with . Note that , , and
Now we use Claim 15 from the Notes on Symmetric Matrices, together with the inequalities
Since , for any , we have , and so
Thus by (1) we have
The same analysis also shows that . Substituting these two bounds into the basic Ahlswede-Winter inequality from the previous lecture, we obtain
Substituting and we get
Multiplying by and using the fact that , we have bounded the probability that any eigenvalue of the sample average matrix is less than or greater than .
Corollary 2 Let be a random, symmetric, positive semi-definite matrix. Define and suppose for some scalar . Let be independent copies of . For any , we have
Proof: Let denote the square root of the pseudoinverse of . Let denote the orthogonal projection on the image of . Define the random, positive semi-definite matrices
Because and , we have . So Claim 16 in Notes on Symmetric Matrices implies
showing that . Next,
So the hypotheses of Theorem 1 are almost satisfied, with the small issue that is not actually the identity, but merely the identity on the image of . But, one may check that the proof of Theorem 1 still goes through as long as every eigenvalue of is either or , i.e., is an orthogonal projection matrix. The details are left as an exercise.