LINEAR STATISTICL MODELS
LINEAR STATISTICL MODELS STAT 714
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This 2 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 714 at University of South Carolina - Columbia taught by I. Dryden in Fall. Since its upload, it has received 45 views. For similar materials see /class/229648/stat-714-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.
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Date Created: 10/26/15
STAT714 LINEAR STATISTICAL MODELS Fall Session 2009 Handout 1 Result M10 Let A be an m gtlt 71 matrix with rA r If A can be partitioned as follows C D A E F v where rA rC T7 and CTXT is nonsingular7 then C 1 0 G o o is a generalized inverse of A Proof Consider CD C 10 CD C D AGAE F 0 0E FHE Em Now since the full matrix A and the subrnatrix C have the same rank7 the rows E F must be linearly dependent on C D7 so that E F KC D for some matrix K Hence E KC7 F KD and so K EC 1 and so F KD EC 1D Therefore7 we have shown AGA A7 7 and so G is a generalized inverse D PTO Result M11 Let Amxm xnxl mel7 and Imm be rnatrices7 and suppose that Ax c is consistent Then7 x is a solution to Ax c if and only if x A c I 7 A Az7 for some 2 E R Proof We know that x A c is a solution Result M9 Suppose that x A c I 7 A Az7 for some 2 E R Thus7 A AA c A 7 AA AZ AA c AXT c that is7 x A c I 7 A Az solves Ax c Conversely7 suppose that x is a solution to Ax c Now7 x A c x 7 A c A cx 7A AX A c I 7 A Ax Thus7 we can write x A c I 7 A Az7 and here we take 2 X For this part of the proof we just need to show that there exists a speci c 2 D Using this result we can vary 2 E R and generate all solutions7 and so we can generate all solutions by just knowing one of them ie7 by knowing A c Note that if A is nonsingular7 A A 1 and x A lc I 7 A lAz A lc ie7 there is just one solution NOTE Consider the general form of the solution to Ax c which is assumed to be consistent ie7 XI A c I 7 A Az We call A c a particular solution The term I 7 A Az is the general solution to the homogeneous equations Ax 07 producing vectors in
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