Question: [e] Let A1, Ag, - - - PA, he in symmetric idempotent matrices and that each 411,- isafsisenxn- HllppnseAl +Ag+---+Am = l- thwthat AiA-j =

$[e] Let A1, Ag, - - - PA,\" he in symmetric$

[e] Let A1, Ag, - - - PA,\" he in symmetric idempotent matrices and that each 411,- isafsisenxn- HllppnseAl +Ag+---+Am = l- thwthat AiA-j = I] whenever 1' 7% j- The-refere- if :r. m lillll'ilrl'ilfl'lJ 1]-l then the cnllectinn ITAII, :T-Agz, - - - , :cT-A-mz are mutually.r independent. This is lrnnwn as flechranls thenrern- Hint: using the spectral dercrnp-nz-iitinn thearern, "i'i'Tl'l-E Ai = {Jill}; WhEl'E.' {11- is a n. x s,- rnatrix; here si dennte at = rklIAI-j- thus L]: is a matrix whese s.- calumru-i are a. subset nf cnlumns 'nrn sc-rne nrtlx'ignnal matrix- Let Q = [til I Q: I I em]- What is QQT ? Hem\" that 3.. = triad; 1What then is 2:1 si '5' Is [1 a. square matrix '3" What then is QTQ "E Additional Hint: see sectinn 3-1-4 {Cachran's thearern} hf A- Agresti \"lendatinns af linear and generalized linear mndels" [available nnline thrcrugh the .lHU library}. PM the applicatinn af {Jnchran's thenrem, see sectien 3-2 f A- Agresti heel: {in piu'ticular sectinn 3-2-2; it is important tr.- nate that the setup in sectian 3-2-1 can easily-r he extended tn multiple nesta'lmedeLs Mnch [Mg C ---C Mg- fnr same K-anddennting by P1. the prajectinn mar-rides assrxaated with model Mh it is trivially true that P1+[PaP1:l+[PaPs}+---+lPKPHJl+IPH=1- What dues Fnchran's thenrem say ah-nut the test statistic {using sum nf squares] fnr mrnparing Mk against Mg far i: a; E 'i'

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