I read about Hadamard gate H and found it's matrix representation as follows:
$$H_1=\frac{1}{\sqrt 2}\begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix}$$
I wanted to know what will be the matrix representation of H in computational basis.
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$\begingroup$ Welcome on the QC SE! We have quite good Latex support here, but we don't really like screenshots. I don't know if your passes well this site, but if it is a border case, your chances are hugely increased by if you use Latex formulas. $\endgroup$– peterhSep 17, 2020 at 15:11
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1 Answer
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This is the matrix representation of $H$ in the computational basis. The first column is the image of $|0\rangle$ and the second column is the image of $|1\rangle$.
The reason that $H$ looks the same in both the computational and the "plus/minus" basis is that $H$ is a self-adjoint (or hermitian) unitary, this makes it very special as it means that its self-inverse since we have that $$H^{−1}=H^\dagger=H\,.$$
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$\begingroup$ So, if this is the matrix representation of H in the computational basis, then what is the matrix representation of H in the hadamard basis? Thanks $\endgroup$– kontamSep 17, 2020 at 15:24
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$\begingroup$ It's just the matrix whose columns are the image of $H|+\rangle$ and $H|-\rangle$ $\endgroup$– CondoSep 17, 2020 at 15:30