Verify matrix is unitary

Hi,
suppose to have an unitary matrix (##U \in \mathbb{C}^{n \times n}##, so that ##U^{\dagger}=U^{-1}##), if you want to verify the unitariety of your matrix, just check if ##UU^{\dagger}= \mathbb{I}##.

 

Are you saying (in the attached file) that ##\begin{pmatrix}i & 0\\ 0 & 1\end{pmatrix}## is the inverse of ##\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}##? It's not. To see this, just multiply these two matrices together.

 

You have at one point, that the "transpose conjugate" of [itex]\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}[/itex] is [itex]\begin{bmatrix}1 & 0 \\ 0 & -i\end{bmatrix}[/itex]. That is correct and that is the inverse matrix.

Below that, you have "inverse matrix" and [itex]\begin{bmatrix} i & 0 \\ 0 & 1\end{bmatrix}[/itex]. I don't know where that came from!

 

What product? The product of the two matrices in post #5 is ##\begin{pmatrix}i & 0 \\ 0 & i\end{pmatrix}##. As HallsofIvy said, the inverse of ##\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}## is ##\begin{pmatrix}1 & 0\\ 0 & -i\end{pmatrix}##. The product of these two matrices is the identity matrix.