da anonymous_0b37e9 » 22/04/2017, 19:09
Il modo più immediato per determinare $g$ rispetto alla base naturale è quello che ti ho proposto nel primo messaggio, dovendo soltanto normalizzare i due vettori ortogonali della consegna e non essendo necessario determinare esplicitamente $U^\bot$:
$((sqrt2/2),(0),(0),(sqrt2/2))((sqrt2/2,0,0,sqrt2/2))+((0),(sqrt2/2),(sqrt2/2),(0))((0,sqrt2/2,sqrt2/2,0))=((1/2,0,0,1/2),(0,1/2,1/2,0),(0,1/2,1/2,0),(1/2,0,0,1/2))$
Tuttavia, si può anche procedere determinando una base ortonormale completa mediante i seguenti due vettori ortonormali appartenenti a $U^\bot$:
$((0),(sqrt2/2),(-sqrt2/2),(0)) ^^ ((sqrt2/2),(0),(0),(-sqrt2/2))$
e ricordando che, per definizione:
$g((sqrt2/2),(0),(0),(sqrt2/2))=((sqrt2/2),(0),(0),(sqrt2/2)) ^^ g((0),(sqrt2/2),(sqrt2/2),(0))=((0),(sqrt2/2),(sqrt2/2),(0)) ^^ g((0),(sqrt2/2),(-sqrt2/2),(0))=((0),(0),(0),(0)) ^^ g((sqrt2/2),(0),(0),(-sqrt2/2))=((0),(0),(0),(0))$
Così, rispetto alla base ortonormale di cui sopra:
$g=((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,0,0,0))$
mentre, rispetto alla base naturale, mediante un cambiamento di base $B^(-1)=B^t=B$:
$g=((sqrt2/2,0,0,sqrt2/2),(0,sqrt2/2,sqrt2/2,0),(0,sqrt2/2,-sqrt2/2,0),(sqrt2/2,0,0,-sqrt2/2))((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,0,0,0))((sqrt2/2,0,0,sqrt2/2),(0,sqrt2/2,sqrt2/2,0),(0,sqrt2/2,-sqrt2/2,0),(sqrt2/2,0,0,-sqrt2/2))^(-1)=((1/2,0,0,1/2),(0,1/2,1/2,0),(0,1/2,1/2,0),(1/2,0,0,1/2))$
Per quanto riguarda $f$, ricordando che, per definizione:
$f((sqrt2/2),(0),(0),(sqrt2/2))=((sqrt2/2),(0),(0),(sqrt2/2)) ^^ f((0),(sqrt2/2),(sqrt2/2),(0))=((0),(sqrt2/2),(sqrt2/2),(0))$
$f((0),(sqrt2/2),(-sqrt2/2),(0))=-((0),(sqrt2/2),(-sqrt2/2),(0)) ^^ f((sqrt2/2),(0),(0),(-sqrt2/2))=-((sqrt2/2),(0),(0),(-sqrt2/2))$
rispetto alla base ortonormale di cui sopra:
$f=((1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1))$
mentre, rispetto alla base naturale:
$f=((sqrt2/2,0,0,sqrt2/2),(0,sqrt2/2,sqrt2/2,0),(0,sqrt2/2,-sqrt2/2,0),(sqrt2/2,0,0,-sqrt2/2))((1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1))((sqrt2/2,0,0,sqrt2/2),(0,sqrt2/2,sqrt2/2,0),(0,sqrt2/2,-sqrt2/2,0),(sqrt2/2,0,0,-sqrt2/2))^(-1)=((0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))$
Ad ogni modo, mediante la formula $[f=2g-Id]$ è senz'altro più immediato.