**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

Finally, since , we get

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

We would like to use Theorem 1 to obtain our desired bound. We just need to check that the hypotheses of the theorem are satisfied. By Fact 6 from the Notes on Symmetric Matrices, we have

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.

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