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04.2.1. A Range Equality for Block Matrices with Orthogonal Projectors

Published online by Cambridge University Press:  11 June 2004

Yongge Tian
Yongge Tian
Affiliation:
Queen's University, Kingston, Canada
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Extract

When considering the general Gauss–Markov model Y = Xβ + e, where E(e) = 0, Cov(e) = V, one often needs to know whether the range inclusion range(X) ⊆ range(V) holds.

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Type
PROBLEMS AND SOLUTIONS
Information
Econometric Theory , Volume 20 , Issue 2 , April 2004, pp. 427
Copyright
© 2004 Cambridge University Press

When considering the general Gauss–Markov model Y = Xβ + e, where E(e) = 0, Cov(e) = V, one often needs to know whether the range inclusion range(X) ⊆ range(V) holds. Show that if V is the following tridiagonal block matrix:

where P and Q are two orthogonal projectors of the same size, then