By George A. F. Seber
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Extra info for A Matrix Handbook for Statisticians (Wiley Series in Probability and Statistics)
Suppose that A = C,=lA,, where each matrix is m x n. We say rank A,. 25. Let A and B be nonnull m x n matrices over F with respective ranks s. If any one of the following conditions hold, then they all hold. , rank additivity). T and 42 RANK (2) There exist nonsingular matrices F and G such that A = F ( I,0 0 0 ) G 0 0 and B = F ( O0 I, 0 0 0 0)G. 0 0 The above matrices are partitioned in the same way, and the bordering zero matrices are of appropriate size; some of the latter matrices are absent if A B has full rank.
A) If r an k A = 0, then A = 0 . This is a simple but key result that can be used to prove the equality of two matrices. (b) If r an k A = 1, then there exist nonzero a and b such that A = ab‘. 5. (Full-Rank Factorization) Any m x n real or complex matrix A of rank r ( T > 0) can be expressed in the form A,,, = CmXTRTXn, where C and R have (full) rank r . We call this a full-rank factorization. The columns of C may be an arbitrary basis of C(A), and then R is uniquely determined, or else the rows of R may be an arbitrary basis of C(A’), and then C is uniquely determined.
37. Ben-Israel and Greville [2003: 121, Rao and Bhimasankaram [2000: 2691, and Seber and Lee [2003: 477, real case]. 38. Graybill [1983: 901 and Harville [1997: 301. 39. Quoted by Kollo and von Rosen [2005: 491. For (a) we first have C(A’B) & C(A’). , C(A’) & C(A’B). The rest are straightforward. 19. A square matrix P such that P2 = P is said to be idempotent. In this section we focus on the geometrical properties of such matrices, which are used extensively in statistics. 6. 20. , V1 n V2 = 0).