  For a singular probability measure $\mu$ on the circle, we show the existence
of positive matrices on the unit disc which admit a boundary representation on
the unit circle with respect to $\mu$. These positive matrices are constructed
in several different ways using the Kaczmarz algorithm. Some of these positive
matrices correspond to the projection of the Szeg\H{o} kernel on the disc to
certain subspaces of the Hardy space corresponding to the normalized Cauchy
transform of $\mu$. Other positive matrices are obtained which correspond to
subspaces of the Hardy space after a renormalization, and so are not
projections of the Szeg\H{o} kernel. We show that these positive matrices are a
generalization of a spectrum or Fourier frame for $\mu$, and the existence of
such a positive matrix does not require $\mu$ to be spectral.
