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edited Jan 10 2012 at 16:14

Noam D. Elkies
18k●4●62●90

Matrix inequality (A-B)^2 <= $(A-B)^2 \leq c (A+B)^2 A+B)^2$ ?

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asked Jan 10 2012 at 4:12

Omar
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Matrix inequality (A-B)^2 <= c (A+B)^2 ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.

My question is: Is there a constant c (independent of A and B and the dimension) such that

$$(A-B)^2 \leq c (A+B)^2?$$

Thanks.