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Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:

$$Tr\left(\frac{1}{1-AA^T}\right)Tr\left(\frac{1}{1-BB^T}\right)-\left(Tr\left(\frac{1}{1-AB^T}\right)\right)^2 $$

$$ \ge Tr\left(\frac{1}{1-AA^T}\right)Tr\left([(A-B)^TC^2(A-B)]\right)\tag{1} $$

This is an attempt to strengthen this inequality: A similar Cauchy-Schwarz inequality with linear-algebra.

I have two questions:

(1) How can one prove this inequality?

(2) Does equality hold if and only if $A=B$?

Improve this question
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    $\begingroup$ Is there any motivation for this inequality? $\endgroup$
    – Russel
    Commented Jun 22, 2014 at 1:34
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    $\begingroup$ For the scalar case, this reduces (as expected) to AM-GM; but after that, this is a Hua-type trace inequality, and seems not to easy to prove..... $\endgroup$
    – Suvrit
    Commented Jun 22, 2014 at 2:33

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