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.
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© 2011 Springer-Verlag Berlin Heidelberg
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Puntanen, S., Styan, G.P.H., Isotalo, J. (2011). Block-Diagonalization and the Schur Complement. In: Matrix Tricks for Linear Statistical Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10473-2_14
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DOI: https://doi.org/10.1007/978-3-642-10473-2_14
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-10473-2
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