By George A. F. Seber
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Additional info for A Matrix Handbook for Statisticians (Wiley Series in Probability and Statistics)
Note that C has a left inverse, namely (C’C)-’C’, and R has a right inverse, R’(RR’)-’. Two full-rank factorizations can be obtained from the singular value decomposition of A (cf. 34e). 6. If A and B are m x n matrices, then r ankA = rankB if and only if there exist a nonsingular m x m matrix C and an n x n nonsingular matrix D such that A = CBD. 7. If C(B) = C(C), then rank(AB) = rank(AC) for all A . 8. 10) and rank(AV) = rank(AR) for all A . Proofs. 1. 1. Abadir and Magnus [2005: 77-78]. 2. (a) and (c) follow from the definition; for (b) see Meyer [2000a: 2151.
Then: (a) S ( A ) is a vector space (even though A may not be). (b) A C S(A). Also S ( A ) is the smallest subspace of V containing A in the sense that every subspace of V containing A also contains S(A). (c) A is a vector space if and only if A = S ( A ) . (4 S[S(A)I = S ( A ) . (e) If A C B,then S( A ) C S ( B ) . (f) S ( A )u S ( B )c S ( Au B ) . (g) S(A n B ) c S ( A )n w). 8. A set of vectors vi (i = 1,2, . . ,r ) in a vector space are linearly aivi = 0 implies that a1 = a2 = . . = a, = 0.
Conversely, if v1 E V1, then C C(P); (b) is similar. 43. Meyer [2000a: 6341. 21. Suppose U has an inner product (,), and let V be a vector subspace with orthogonal complement V I , namely V' Then U E v}. V' so that every v E U can be expressed uniquely in the form The vectors v1 and v2 are called the orthogoad projections of v onto V and V', respectively (we shall omit the words "along V'" and "along V" , respectively). Orthogonal projections will, of course, share the same properties as general projections.
A Matrix Handbook for Statisticians (Wiley Series in Probability and Statistics) by George A. F. Seber