Properties of matrix of linear transformation w.r.t an orthonormal basis

Question

Suppose $V$ is a finite dimensional Inner Product Space and $T : V \to V$ be linear operator. I just wondered what can be some special properties of matrix of $T$ w.r.t an orthonormal basis.
I cannot see it clearly if any. Is it orthogonal. If so how can we prove it?? Kindly help! Thanks and regards

Answer 1

A useful property is that the matrix of the adjoint $T^*$ w.r.t an orthonormal basis $E$ is given by the conjugate transpose of the matrix of $T$, i.e. $$(T^*)_{(E)} = (T_{(E)})^*$$

From there you can infer:

• $T$ is normal if and only if $T_{(E)}$ is a normal matrix
• $T$ is self-adjoint if and only if $T_{(E)}$ is a hermitian matrix
• $T$ is unitary if and only if $T_{(E)}$ is a unitary matrix