If matrices $A,B$ and $(A^2-B^2)$ are all positive semidefinite, is $A-B$ also?

Asked May 18 '19 at 09:49

Active May 18 '19 at 09:53

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The question is in the title. I have not found a counter example, so I try to prove it. However, contract diagonalization at the same time seems useless. Thanks very much.

edited May 18 '19 at 09:53

Jean Marie

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asked May 18 '19 at 09:49

zxcvbn6666

Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos May 18 '19 at 09:51
Sorry，I forgot this .A,B are all real and symmetrical. – zxcvbn6666 May 18 '19 at 10:43

If matrices $A,B$ and $(A^2-B^2)$ are all positive semidefinite, is $A-B$ also?

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