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EnvivyEvoxys6 2022-06-30 Answered
Let be a norm invariant under unitary trasformations.
Is it true that
( L ^ L G )
is minimized when L ^ = L ( L ^ and G are fixed, L is the only variable quantity), i.e ( L ^ L G ) = ( 0 G ) ?
I'm unable to prove using just elementar inequalities such as triangle inequality or norm properties, any help would be appreciated.
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Answers (1)

Kroatujon3
Answered 2022-07-01 Author has 19 answers
If the transformation U that maps [ a , b ] T into [ a , b ] T is unitary (it might depend on your exact setting), then :
x + U x 2 1 2 ( x + U x ) = x
but if x = [ a , b ] T then x + U x 2 = [ 0 , b ] T .
Therefore min a [ a , b ] T = [ 0 , b ] T . This can then be adapted by translation to your setting.
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