If $|([S]_B)_ij|=|([S']_B)_ij|=delta_ij$ is there a basis $B''$ s.t. $|([S'circ S]_B'')_ij|=delta_ij$

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Let $V$ a $mathbb C-$vector space of finite dimension. Let $S,S'in mathcal L(V)$. If there are orthonormal basis $B,B'$ of $V$ s.t. $$|([S]_B)_ij|=|([S']_B)_ij|=delta_ij$$ is there an orthonormal basis $B''$ of $V$ s.t. $$|([S'circ S]_B'')_ij|=delta_ij ?$$



My principal problem is to understand $|([S]_B)_ij|$, what does it mean ? I know that $[S]_BB$ is a matrix. That $[S]_BB[v]_B$ is a vector, and thus $|[S]_BB[v]_B|=sqrtleft<[S]_BB[v]_B,[S]_BB[v]_Bright>$ make sense (even if not any scalar product is given). But,



1) What would be $([S]_B)_ij$ ?



2) And then, what would be $|([S]_B)_ij|$ ?




linear-algebra




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  • Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
    – xbh
    Jul 29 at 13:28














up vote
0
down vote

favorite












Let $V$ a $mathbb C-$vector space of finite dimension. Let $S,S'in mathcal L(V)$. If there are orthonormal basis $B,B'$ of $V$ s.t. $$|([S]_B)_ij|=|([S']_B)_ij|=delta_ij$$ is there an orthonormal basis $B''$ of $V$ s.t. $$|([S'circ S]_B'')_ij|=delta_ij ?$$



My principal problem is to understand $|([S]_B)_ij|$, what does it mean ? I know that $[S]_BB$ is a matrix. That $[S]_BB[v]_B$ is a vector, and thus $|[S]_BB[v]_B|=sqrtleft<[S]_BB[v]_B,[S]_BB[v]_Bright>$ make sense (even if not any scalar product is given). But,



1) What would be $([S]_B)_ij$ ?



2) And then, what would be $|([S]_B)_ij|$ ?




linear-algebra




share|cite|improve this question



















  • Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
    – xbh
    Jul 29 at 13:28












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $V$ a $mathbb C-$vector space of finite dimension. Let $S,S'in mathcal L(V)$. If there are orthonormal basis $B,B'$ of $V$ s.t. $$|([S]_B)_ij|=|([S']_B)_ij|=delta_ij$$ is there an orthonormal basis $B''$ of $V$ s.t. $$|([S'circ S]_B'')_ij|=delta_ij ?$$



My principal problem is to understand $|([S]_B)_ij|$, what does it mean ? I know that $[S]_BB$ is a matrix. That $[S]_BB[v]_B$ is a vector, and thus $|[S]_BB[v]_B|=sqrtleft<[S]_BB[v]_B,[S]_BB[v]_Bright>$ make sense (even if not any scalar product is given). But,



1) What would be $([S]_B)_ij$ ?



2) And then, what would be $|([S]_B)_ij|$ ?




linear-algebra




share|cite|improve this question











Let $V$ a $mathbb C-$vector space of finite dimension. Let $S,S'in mathcal L(V)$. If there are orthonormal basis $B,B'$ of $V$ s.t. $$|([S]_B)_ij|=|([S']_B)_ij|=delta_ij$$ is there an orthonormal basis $B''$ of $V$ s.t. $$|([S'circ S]_B'')_ij|=delta_ij ?$$



My principal problem is to understand $|([S]_B)_ij|$, what does it mean ? I know that $[S]_BB$ is a matrix. That $[S]_BB[v]_B$ is a vector, and thus $|[S]_BB[v]_B|=sqrtleft<[S]_BB[v]_B,[S]_BB[v]_Bright>$ make sense (even if not any scalar product is given). But,



1) What would be $([S]_B)_ij$ ?



2) And then, what would be $|([S]_B)_ij|$ ?





linear-algebra





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 29 at 10:20









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  • Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
    – xbh
    Jul 29 at 13:28
















  • Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
    – xbh
    Jul 29 at 13:28















Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
– xbh
Jul 29 at 13:28




Maybe you should refer back to your textbook? The "norm" is confusing for us, too…
– xbh
Jul 29 at 13:28















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