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HW8_prob1_GPS1_G1

# HW8_prob1_GPS1_G1 - that S is orthogonal Start with S − 1...

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Problem Goldstein 4-1 (3 rd ed. 4.1) Associativity Say we have three matrices, P , Q , R , whose dimensions are such that the matrix multiplications PQ = S , (1) and QR = T , (2) are de fi ned. Then, to prove associativity, we need to show that SR = PT . (3) Consider the ij element of each side of Eq.(3), ( SR ) ij = X k S ik R kj , (4) ( PT ) ij = X k P ik T kj . (5) From Eqs.(1) and (2), we have S ik = ( PQ ) ik = X l P il Q lk , (6) and T kj = ( QR ) kj = X l Q kl R lj . (7) Next we substitute Eq.(6) into Eq.(4) to obtain ( SR ) ij = X k X l P il Q lk R kj , (8) and we substitute Eq.(7) into Eq.(5) to obtain ( PT ) ij = X k P ik X l Q kl R lj = X k X l P ik Q kl R lj . (9) In Eq.(8), we can invert the order of summation to get ( SR ) ij = X l X k P il Q lk R kj , (10) 1

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and then we can switch the dummy summation indices l and k , i.e., let l equal k and k equal l , to get ( SR ) ij = X k X l P ik Q kl R lj , (11) which is clearly identical to Eq.(9). Thus, Eq.(3) is proved. Orthogonality Now, assume that P and Q are orthogonal, i.e., that P 1 = P T and Q 1 = Q T , where superscript T denotes the transpose of a matrix. We want to prove
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Unformatted text preview: that S is orthogonal. Start with S − 1 , S − 1 = ( PQ ) − 1 = Q − 1 P − 1 , (12) where we have used the well-known property that the inverse of a matrix product is the product of the inverse matrices in reverse order. Next, we replace Q − 1 and P − 1 by their respective transposes to get S − 1 = Q − 1 P − 1 = Q T P T = ( PQ ) T , (13) where the last equality follows from another well-known property that the trans-pose of a matrix product is the product of the transposed matrices in reverse order. Finally, we use Eq.(1) to replace PQ in Eq.(13), leaving us with S − 1 = S T , (14) which establishes the desired result. 2...
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