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## Unitary matrix and matrix inequality [closed]

Dear all,

Suppose U and V are unitary matrix, A and B are positive definite,

Does:

$UAU^{-1} < VBV^{-1}$

implies $A< B$

and vice versa?

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 What do you mean with $A 0 or (B-A) is positive definite. U and V not necessarily diagonalize A and B, just to make them block diagonal. Thanks. – hayu Aug 16 at 15:59 ## closed as too localized by Deane Yang, Bill Johnson, Mark Sapir, suVRit, Mark MeckesAug 16 at 17:02 ## 1 Answer No: think of the case of$A$,$B$diagonal;$U$identity, and$V\$ a permutation operator.

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 thank you ^_^ – hayu Aug 16 at 16:18