Orthonormal basis for Hermitian matrix

Question

Suppose there is a hermitian matrix.

Then, Can we always find out orthonormal basis for this matrix ?

And, Is there any relationship between hermitian matrix and hermitian transformation?

If matrix is hermitian, does that imply transformation is hermitian?

or

If the transformation is hermitian, then matrix of T is hermitian?

Answer 1

Yes; it is indeed true that a transformation if and only if its matrix is Hermitian (with respect to any orthonormal basis).

Proof: Let $A$ be the standard matrix associated with $T$.

Suppose that $A$ is Hermitian. Then for all $x,y$, $$ \langle T(x),y \rangle = y^* (Ax) = (y^* A^*)x = (Ay)^*x = \langle x,T(y) \rangle $$ as desired.

Suppose $T$ is Hermitian. Then for all $x,y$, we have $$ y^*Ax = \langle T(x),y \rangle = \langle x,T(y)\rangle = (Ay)^*x = y^*A^* x $$ It follows that $A = A^*$

Furthermore, by the spectral theorem, every Hermitian matrix has an orthonormal eigenbasis.