Linear Algebra and Linear Models by R. B. Bapat

Author:R. B. Bapat
Language: eng
Format: epub, pdf
Publisher: Springer London, London

We now prove a matrix theoretic formulation of one version of Cochran’s Theorem.

8.6

Let A 1,…,A k be n×n matrices with . Then the following conditions are equivalent: (i)

(ii) , i=1,…,k

(iii) A i A j =0, i≠j.

Proof

(i) ⇒ (iii). Let A i =B i C i be a rank factorization, i=1,…,k. Then

and hence

Since , [B 1⋯B k ] is a square matrix and therefore

Thus C i B j =0, i≠j. It follows that for i≠j,

(iii) ⇒ (ii). Since ,

It follows that .

(ii) ⇒ (i). Since A i is idempotent, . Now

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