If we consider some unitary operator in matrix form given by $

U = \left[ {\begin{array}{cc}

P_{00} & P_{01} \cdot \cdot & P_{0J} \\

P_{10} & \\

\cdot &\\

\cdot\\

P_{J0} \cdot\cdot\cdot& & P_{JJ} \\

\end{array} } \right]$

Where each $P_{nm}$ are matrices (sub-blocks of $U$). How can we show that the first $J$ columns of $U$ are orthonormal vectors?

Please see my proposed answer let me know what you think about the validity. Proposed answer:

Let $U_i$ be the i-th column of $U$ and consider the elements of $U^{\dagger}U$ then: $$[U^{\dagger}U]_{ij} = \sum_{k}P_{ik}^{\dagger}P_{kj} = U_{i}^{\dagger}U_{j} = \delta_{ij} := \begin{cases} I~~~\text{if }i=j\\ \mathbf{0}~~~\text{if } i \neq j \end{cases}$$

Where $I$ is the identity matrix and $0$ is the zero matrix.

This is just the inner-product $\langle U_i, U_j \rangle$ (if we defined the inner product in an analogous way for column vectors as we usually do) and the norm $\langle U_i, U_i \rangle = I := || U_i ||^2$.

Thanks for any help.