Block-Diagonalization and the Schur Complement

Simo Puntanen (), George P. H. Styan () and Jarkko Isotalo ()
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Simo Puntanen: University of Tampere, School of Information Sciences
George P. H. Styan: McGill University, Department of Mathematics & Statistics
Jarkko Isotalo: University of Tampere, School of Information Sciences

Chapter Chapter 13 in Matrix Tricks for Linear Statistical Models, 2011, pp 291-304 from Springer

Abstract: Abstract In this chapter we present a block-diagonalization result for a symmetric nonnegative definite matrix. We may emphasize that the block-diagonalization result, sometimes called the Aitken block-diagonalization formula, due to Aitken (1939, Ch. 3, §29), is mathematically indeed quite simple just as it is. However, it is exceptionally handy and powerful tool for various situations arising in linear models and multivariate analysis, see, e.g., the derivation of the conditional multinormal distribution in Anderson (2003, §2.5); cf. also (9.21)–(9.22) (p. 193). We also consider the Schur complements whose usefulness in linear models and related areas can hardly be overestimated.

Keywords: Random Vector; Generalize Inverse; Penrose Inverse; Lecture Theatre; Nonnegative Definite Matrix (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-10473-2_14

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