Question

A hermitian matrix remains hermitian under unitary transformation

Answer 1

matrix is hermitian if A∗=AA∗=A where A∗A∗ is the conjugated and transposed of AA. Unitary matrices have the property that U⋅U∗=IdU⋅U∗=Id where Id is the identity. So in special we have U∗=U−1U∗=U−1.

Now we look at the transformed hermitian:
U−1AU=U∗AU
U−1AU=U∗AU
if conjugate and transpose this we have
(U∗AU)∗=U∗A∗(U∗)∗=U∗A∗U=U∗AU
(U∗AU)∗=U∗A∗(U∗)∗=U∗A∗U=U∗AU
which says that the transformed still hermitian?

Answer 2

A matrix is hermitian if A∗=A where A∗ is the conjugated and transposed of A. Unitary matrices have the property that U⋅U∗=Id where Id is the identity. So in special we have U∗=U−1.

Now we look at the transformed hermitian:

U−1AU=U∗AU

if conjugate and transpose this we have

(U∗AU)∗=U∗A∗(U∗)∗=U∗A∗U=U∗AU

which says that the transformed still hermitian.