I was trying to prove if the Hadamard gate matrix is unitary. Maybe you can help me out in figuring out the conjugate tranpose of this matrix:
$\frac{1}{\surd2} \begin{bmatrix}1 & 1 \\1 & -1 \end{bmatrix}$
My thoughts so far
I suspect that the conjugate transpose is
$\frac{1}{\surd2} \begin{bmatrix}-1 & -1 \\-1 & 1 \end{bmatrix}$. If this is right, then multiplying this adjoint to the actual matrix, will result in this matrix
$\begin{bmatrix}-1 & 0 \\0 & -1 \end{bmatrix}$. Is this the same as the identity matrix?
Best Answer
This matrix is symmetric and all of its entries are real, so it's equal to its conjugate transpose.
The matrix you are asking about is different from the identity matrix.
But the original matrix is unitary.