Hermitian Matrix

Let a “Hermitian matrix” be a matrix that equals its own Conjugate transpose

If and only if AA is a matrix and is Hermitian, then this is true

A=A+\begin{gather*} A=A^+ \end{gather*}

This means a bra and ket are Hermitian Matrix of each other!

Another example:

∣ψ⟩=[ab]∣ϕ⟩=[cd]⇒⟨ϕ∣=[c∗d∗]\begin{gather*} \left\lvert \psi \right\rangle=\begin{bmatrix}a\\b\end{bmatrix}\quad \left\lvert \phi \right\rangle=\begin{bmatrix}c\\d\end{bmatrix}\Rightarrow\left\langle \phi \right\rvert=\begin{bmatrix}c^* & d^*\end{bmatrix} \end{gather*}