A Canonical Representation of Block Matrices with Applications to Covariance and Correlation Matrices

TL;DR

This paper introduces a canonical form for block matrices that simplifies computations like determinant, inverse, and matrix functions, significantly aiding the analysis of covariance and correlation matrices in large datasets.

Contribution

The paper presents a new canonical representation for block matrices, enabling easier computation and analysis, especially for large covariance and correlation matrices.

Findings

Simplifies computation of determinants, inverses, and matrix functions.

Facilitates regularization and testing of block structures in large matrices.

Demonstrates practical utility with an empirical asset returns dataset.

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 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.