An orthogonal projection is an Hermitian matrix $P$ such that $P^2=P$. Denote $U^*$ the conjugate transpose of a matrix $U$.

It can be easily shown that for two projections $P_1$ and $P_2$, there exists a unitary $U$ such that both $UP_1 U^*$ and $UP_2U^*$ are block diagonal with blocks of size one or two (And both resulting matrices have the same block structure).

My question is whether this block decomposition of projections can be generalised, for more than two projections: Given orthogonal projections $P_1, P_2, ..., P_k$, Is there a unitary $U$ such that for each $i$, $UP_i U^*$ is block diagonal with blocks of size at most $k$? (The resulting matrices must have the same block structure)

Two weaker questions are:

Is there a bound on the size of the blocks in function of $k$ only, i.e. in function of the number of projectors independently of their dimensions?

If a block decomposition is not possible, then what about decomposing the projectors into $k$-diagonal matrices? (All entries of the matrix are zero except (possible) for the diagonal and the $k$-upper and $k$-lower diagonals)

I would deeply appreciate any help or reference on how to handle these problems.

Best regards,

Mateus