# What is the most general unitary that commutes with a one dimensional projector in a finite dimensional Vector Space

Given a vector space of finite dimension $N$ with an orthonormal basis $\mathcal{B}=\{|0\rangle,\ldots,|N-1\rangle \}$ what is the most general unitary operation that commutes with the projector onto one of the basis elements (say $|0\rangle$), i.e., what is the most general $\mathcal{U}$ such that $\mathcal{U}^\dagger \mathcal{U}=\mathcal{U} \mathcal{U}^\dagger=\mathbb{I}$ and $\left[\mathcal{U},|0\rangle \langle 0| \right]\equiv \mathcal{U} |0\rangle \langle 0|- |0\rangle \langle 0|\mathcal{U}=0$. A partial answer to the question is: \begin{eqnarray*} \mathcal{U(\lambda,\theta,\gamma)}&=&\exp\left(i G(\lambda,\theta,\gamma )\right)\\ G(\lambda,\theta,\gamma)&=&\sum_{k=1}^{N-1}\lambda_k |k \rangle \langle k|\\ &&+\sum_{k=1}^{N-1}\sum_{l>k}^{N-1} \theta_{l,k} \left(|k \rangle \langle l|+|l \rangle \langle k|\right)\\ &&+i\sum_{k=1}^{N-1}\sum_{l>k}^{N-1} \gamma_{l,k} \left(|k \rangle \langle l|-|l \rangle \langle k|\right) \end{eqnarray*} For $N=2$ it is indeed the answer to the question, but I am not sure for higher dimensions.

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I am not familiar with the bra-ket notation. However, since $\mathcal{U}$ commutes with the projector into the span of the first basis vector, the matrix of $\mathcal{U}$ with respect to the given orthonormal basis is a unitary matrix in the form of $\pmatrix{z\\ &V}$, where $z$ is a complex number with unit modulus and $V$ is an $(n-1)\times(n-1)$ unitary matrix.