  We study a relation between brick $n$-tuples of subspaces of a finite
dimensional linear space, and irreducible $n$-tuples of subspaces of a finite
dimensional Hilbert (unitary) space such that a linear combination, with
positive coefficients, of orthogonal projections onto these subspaces equals
the identity operator. We prove that brick systems of one-dimensional subspaces
and the systems obtained from them by applying the Coxeter functors (in
particular, all brick triples and quadruples of subspaces) can be unitarized.
For each brick triple and quadruple of subspaces, we describe sets of
characters that admit a unitarization.
