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HW8_prob2_GPS2_G2

# HW8_prob2_GPS2_G2 - † indicates an adjoint...

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Problem Goldstein 4-2 (3 rd ed. 4.2) Transposition For two matrices, A and B , we want to prove ( AB ) T = B T A T , (1) where superscript T indicates a matrix transpose. Consider the ji element of the left-hand-side ( AB ) T ji = ( AB ) ij . (2) By the rules of matrix multiplication, we have ( AB ) ij = X k A ik B kj . (3) From the de fi nition of the transpose, we may write A ik = ( A T ) ki (4) and B kj = ( B T ) jk . (5) Thus, Eq.(3) may be rewritten as ( AB ) ij = X k ( B T ) jk ( A T ) ki = ( B T A T ) ji , (6) from which Eq.(1) follows. Adjointness Prove that the adjoint of a product of matrices equals the product of the adjoint matries in reverse order, ( AB ) = B A , (7) where superscript
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Unformatted text preview: † indicates an adjoint matrix (transpose plus complex con-jugation). The proof is simpler than the one above. Because the complex conjugate ( ∗ ) of a product is the product of the complex conjugates, we can immediately write B † A † = ( B T ) ∗ ( A T ) ∗ = ( B T A T ) ∗ . (8) Now use the theorem just proven above to rewrite Eq.(8) further as B † A † = ( B T A T ) ∗ = (( AB ) T ) ∗ = ( AB ) † , (9) which is the desired result. 1...
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