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 Saywehavethreematr ices , P , Q , R , whose dimensions are such that the matrix multiplications PQ = S , (1) and QR = T , (2) are de f 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 ,

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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 ,toget ( 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 ,wheresuperscr ipt T
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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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