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AppendixAProofofTheorem1ProofLetU22Rmm V22Rnn P22R qandQ22RkkqbecolumnorthogonalmatricessuchthatUU1U2VV1V2PP1P2andQQ1Q2areorthogonalrespectivelyThenitisobvioustha

AppendixA.ProofofTheorem1Proof.LetU22Rm(m ) V22Rn(n ) P22R ( q)andQ22Rk(kq)becolumnorthogonalmatricessuchthatU=U1U2 V=V1V2 P=P1P2andQ=Q1Q2areorthogonal respectively.Then itisobvioustha

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