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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. $\endgroup$ – peterh - Reinstate Monica Sep 17 at 15:11
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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 $\endgroup$ – kontam Sep 17 at 15:24
  • $\begingroup$ It's just the matrix whose columns are the image of $H|+\rangle$ and $H|-\rangle$ $\endgroup$ – Condo Sep 17 at 15:30

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