{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW8_prob2_GPS2_G2 - † indicates an adjoint...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern