- Hermitian Adjoints
S0 4106S
Calculate the following quantities for the matrices:
\[A\doteq
\begin{pmatrix}
1&0&0\\ 0&0&1\\ 0&-1&0\\
\end{pmatrix}
\hspace{2em}
B\doteq
\begin{pmatrix}
a&b&c\\ d&e&f\\ g&h&j\\
\end{pmatrix}
\hspace{2em}
C\doteq
\begin{pmatrix}
\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\\
\end{pmatrix}
\]
and the vectors:
\[\left|D\right\rangle\doteq
\begin{pmatrix}
1\\ i\\ -1\\
\end{pmatrix}
\hspace{2em}
\left|E\right\rangle\doteq
\begin{pmatrix}
1\\ i\\
\end{pmatrix}
\hspace{2em}
\left|F\right\rangle\doteq
\begin{pmatrix}
1\\ -1\\
\end{pmatrix}
\]
- \(A^{\dagger}\)
- \(\vert E\rangle^{\dagger}\equiv\langle E\vert\)
- \(\langle D\vert A\vert D\rangle\)
- \(\left(A\vert D\rangle\right)^{\dagger}\)
- Using explicit matrix multiplication (without using a theorem)
verify that \(\left(A\vert D\rangle\right)^{\dagger}
=\langle D\vert
A^{\dagger}\)