**Authors:**Ilya Archakov (University of Vienna) and Peter Hansen (University of North Carolina at Chapel Hill)

**Abstract:**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 matrices and block correlation matrices, where evaluation of the Gaussian log-likelihood and estimation is 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 and to test block structures in matrices.

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