  We obtain a canonical representation for block matrices. The representation
facilitates simple computation of the determinant, the matrix inverse, and
other powers of a block matrix, as well as the matrix logarithm and the matrix
exponential. These results are particularly useful for block covariance and
block correlation matrices, where evaluation of the Gaussian log-likelihood and
estimation are greatly simplified. We illustrate this with an empirical
application using a large panel of daily asset returns. Moreover, the
representation paves new ways to regularizing large covariance/correlation
matrices, test block structures in matrices, and estimate regressions with many
variables.
