Adjoint Operator $g=ff^{*}+f^{*}f$
$begingroup$
Let V be a finite Vector Space over $mathbb{R}$. Let $phi$ be a inner product positive-definite. Let $f$ be an endomorphism of V we define $f^{*}$ as its adjoint.
Now we define $g=ff^{*}+f^{*}f$.
Show that $ker g= ker f cap kerf^{*}$
$supseteq$) Let $v in ker f cap kerf^{*}$, $g(v)=ff^{*}(v)+f^{*}f(v)=0+0=0$.
Now I'm stuck with the other inclusion, any hint?
Here what I've tried so far, it might be wrong:
Let $v in ker g$ it means that $ff^{*}(v)=-f^{*}f(v)$ then:
$phi(g(v),g(v))=0 implies phi(ff^{*}(v)+f^{*}f(v),ff^{*}(v)+f^{*}f(v))=0 implies phi(ff^{*}(v),ff^{*}(v))+2phi(ff^{*}(v),f^{*}f(v))+phi(f^{*}f(v),f^{*}f(v)) $ Using the aforementioned relation I get: $phi(ff^{*}(v),ff^{*}(v))+phi(f^{*}f(v),f^{*}f(v))-2phi(ff^{*}(v),ff^{*}(v))=0 implies phi(f^{*}f(v),f^{*}f(v))-phi(ff^{*}(v),ff^{*}(v))=0 $ I feel like i should use the properties of the adjoint "erasing" it in the inner product but i could not work out the right proof.
Thanks in advance
linear-algebra
asked Jan 6 at 12:43
SMCSMC
6010
$endgroup$
add a comment |
$begingroup$
Let V be a finite Vector Space over $mathbb{R}$. Let $phi$ be a inner product positive-definite. Let $f$ be an endomorphism of V we define $f^{*}$ as its adjoint.
Now we define $g=ff^{*}+f^{*}f$.
Show that $ker g= ker f cap kerf^{*}$
$supseteq$) Let $v in ker f cap kerf^{*}$, $g(v)=ff^{*}(v)+f^{*}f(v)=0+0=0$.
Now I'm stuck with the other inclusion, any hint?
Here what I've tried so far, it might be wrong:
Let $v in ker g$ it means that $ff^{*}(v)=-f^{*}f(v)$ then:
$phi(g(v),g(v))=0 implies phi(ff^{*}(v)+f^{*}f(v),ff^{*}(v)+f^{*}f(v))=0 implies phi(ff^{*}(v),ff^{*}(v))+2phi(ff^{*}(v),f^{*}f(v))+phi(f^{*}f(v),f^{*}f(v)) $ Using the aforementioned relation I get: $phi(ff^{*}(v),ff^{*}(v))+phi(f^{*}f(v),f^{*}f(v))-2phi(ff^{*}(v),ff^{*}(v))=0 implies phi(f^{*}f(v),f^{*}f(v))-phi(ff^{*}(v),ff^{*}(v))=0 $ I feel like i should use the properties of the adjoint "erasing" it in the inner product but i could not work out the right proof.
Thanks in advance
linear-algebra
asked Jan 6 at 12:43
SMCSMC
6010
$endgroup$
add a comment |
$begingroup$
Let V be a finite Vector Space over $mathbb{R}$. Let $phi$ be a inner product positive-definite. Let $f$ be an endomorphism of V we define $f^{*}$ as its adjoint.
Now we define $g=ff^{*}+f^{*}f$.
Show that $ker g= ker f cap kerf^{*}$
$supseteq$) Let $v in ker f cap kerf^{*}$, $g(v)=ff^{*}(v)+f^{*}f(v)=0+0=0$.
Now I'm stuck with the other inclusion, any hint?
Here what I've tried so far, it might be wrong:
Let $v in ker g$ it means that $ff^{*}(v)=-f^{*}f(v)$ then:
$phi(g(v),g(v))=0 implies phi(ff^{*}(v)+f^{*}f(v),ff^{*}(v)+f^{*}f(v))=0 implies phi(ff^{*}(v),ff^{*}(v))+2phi(ff^{*}(v),f^{*}f(v))+phi(f^{*}f(v),f^{*}f(v)) $ Using the aforementioned relation I get: $phi(ff^{*}(v),ff^{*}(v))+phi(f^{*}f(v),f^{*}f(v))-2phi(ff^{*}(v),ff^{*}(v))=0 implies phi(f^{*}f(v),f^{*}f(v))-phi(ff^{*}(v),ff^{*}(v))=0 $ I feel like i should use the properties of the adjoint "erasing" it in the inner product but i could not work out the right proof.
Thanks in advance
linear-algebra
asked Jan 6 at 12:43
SMCSMC
6010
$endgroup$
Let V be a finite Vector Space over $mathbb{R}$. Let $phi$ be a inner product positive-definite. Let $f$ be an endomorphism of V we define $f^{*}$ as its adjoint.
Now we define $g=ff^{*}+f^{*}f$.
Show that $ker g= ker f cap kerf^{*}$
$supseteq$) Let $v in ker f cap kerf^{*}$, $g(v)=ff^{*}(v)+f^{*}f(v)=0+0=0$.
Now I'm stuck with the other inclusion, any hint?
Here what I've tried so far, it might be wrong:
Let $v in ker g$ it means that $ff^{*}(v)=-f^{*}f(v)$ then:
$phi(g(v),g(v))=0 implies phi(ff^{*}(v)+f^{*}f(v),ff^{*}(v)+f^{*}f(v))=0 implies phi(ff^{*}(v),ff^{*}(v))+2phi(ff^{*}(v),f^{*}f(v))+phi(f^{*}f(v),f^{*}f(v)) $ Using the aforementioned relation I get: $phi(ff^{*}(v),ff^{*}(v))+phi(f^{*}f(v),f^{*}f(v))-2phi(ff^{*}(v),ff^{*}(v))=0 implies phi(f^{*}f(v),f^{*}f(v))-phi(ff^{*}(v),ff^{*}(v))=0 $ I feel like i should use the properties of the adjoint "erasing" it in the inner product but i could not work out the right proof.
Thanks in advance
linear-algebra
linear-algebra
asked Jan 6 at 12:43
SMCSMC
6010
asked Jan 6 at 12:43
SMCSMC
6010
asked Jan 6 at 12:43
SMCSMC
6010
asked Jan 6 at 12:43
SMCSMC
6010
asked Jan 6 at 12:43
SMCSMC
6010
6010
add a comment |
add a comment |
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