I am studying for a final exam and came across a sentence in my linear algebra textbook stating that "a linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an orthonormal basis. "
Could someone clarify why it is sufficient to show normality of a linear operator by showing normality of the matrix representation the linear operator with respect to an orthonormal basis?
Thank you.
An operator $T$ is normal if $TT^*=T^*T$, where $T^*$ is the adjoint operator.
A matrix $M$ is normal if $MM^*=M^*M$, where $M^*$ is the conjugate-transpose matrix.
These are equivalent, because if in some orthonormal basis $T$ is represented by $M$, then $T^*$ is represented by $M^*$. Indeed the $(i,j)$ entry of $M^*$ is $\langle T^*e_j, e_i\rangle $, which is the same as $\overline{\langle Te_i, e_j\rangle} $.