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Your matrix representation is incorrect. In the first case, a $\mathbf H$ gate is applied to the first qubit $A$ and an identity gate $\Bbb I$ is applied to the second qubit $B$. So the effective quan …

Mostly I'm confused over whether the common convention is to use +i or -i along the anti-diagonal of the middle 2x2 block. The former. There are two $+i$'s along the anti-diagonal of the middle …

That's not the right way to look at it. In quantum mechanics, time evolutions are considered to be unitary and any unitary evolution can be written as a sequence of unitary operators $U_1, U_2, U_3,\l …

Consider the linear maps $A: V\to W$ and $B: W\to X$. The composition $BA$ is a linear map from $V$ to $X$. Now, how can $\mathcal{M}(BA)$ be computed from $\mathcal{M}(B)$ and $\mathcal{M}(A)$? $\mat …

A $9\times 9$ matrix $H$ can act on a $9$ dimensional state vector, say something like: $$|\Psi\rangle = a_0|0\rangle + a_1|1\rangle + .... + a_8|8\rangle$$ Now, say you want to find the matrix whi …

Firstly, there's a conceptual error in your calculation of the eigenvectors. $$\begin{bmatrix} x\\y \end{bmatrix}=\begin{bmatrix} \tfrac{1}{\sqrt{2}} &\tfrac{1}{\sqrt{2}} \\ \tfrac{1}{\sqrt{2}} & -\t …