there is A matrices order n over C.we define M_{nxn}^{c} a transformation T as T(B)=AB for every B \in M_{nxn}^{c}
A.prove that if A unitary if and only if T is unitary TT^{*}=T^{*}T=I?
B.prove that if A normal if and only if T is normal (TT^{*}=T^{*}T)?
C.prove that if A close to itselfs if and only if T is close to itself ( T=-T^{*})?
i wrote the definition near each question ,also i know this T*=\overline{T^{t}}.
for A.
T(B)T(B)*=AB(AB)^{*}=AB\overline{AB^{t}}
what next?
the prof uses (T(x1),x2)=(x1, T^{*}(x1))
i dont know this definition and how it linked to the transformation we are asked to do
?