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HermitianSpaces: an Adjoint property proof (help)

  1. Sep 6, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]V[/tex] is a linear space over [tex]C[/tex], finite n-dimensional

    [tex]h: VxV \rightarrow C[/tex] is an Hermitian Product, POSITIVE DEFINED

    and so

    [tex](V,h)[/tex] Hermitian Space
    ---------------------------------------------------------------

    [tex]L: V \rightarrow V[/tex], is a Linear Endomorphism of V

    [tex]L^{*}[/tex] is the ADJOINT of [tex]L[/tex]

    [tex]h(L(v),w) = h(v,L^{*}(w)) \forall v,w\in V[/tex] (adjoint definition)


    -------------------------------------------------------------------------------
    Given:

    [tex]B=\left\{v_{1},v_{2},v_{3},....................,v_{n}\right\}[/tex] is a h-ORTHORNORMAL basis for [tex]V[/tex]

    [tex]M\itshape^{B}_{B}(L)[/tex] i.e. matricial representation "from basis B to basis B" of L

    [tex]M\itshape^{B}_{B}(L^{*})[/tex] i.e. matricial representation "from basis B to basis B" of L* (the adjoint of L)

    I need to proof that:

    [tex]M\itshape^{B}_{B}(L^{*})= (M\itshape^{B}_{B}(L))^{*}[/tex]

    Please help me, I'm not able to proof it. tnx
     
  2. jcsd
  3. Sep 6, 2008 #2

    Dick

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    Start by writing down the expression for the ij_th element of both of those matrices (i.e. what is a general matrix element) in terms of the {v_i}. Now remember '*' of a matrix is the conjugate transpose.
     
  4. Sep 6, 2008 #3
    Tnx for the quick replay, I'll try as you say...
     
  5. Sep 6, 2008 #4
    I solved it !!!!!!!!!!!!!!!!:rofl:

    [tex]B=\left\{v_{1},v_{2},v_{3},....................,v_{n}\right\}[/tex] is a h-ORTHORNORMAL basis for [tex]V[/tex]

    ------------------------------------------------------------------

    [tex]M\itshape^{B}_{B}(L)=(a_{x,y})[/tex] j-th Columns is:

    [tex]L(v_{j})= a_{1j}v_{1} +....+a_{nj}v_{n} [/tex]



    -------------------------------------------------------------------

    [tex]M\itshape^{B}_{B}(L^{*})=(b_{x,y}) [/tex] k-th COLUMN IS:

    [tex]L^{*}(v_{k})= b_{1j}v_{1}+....+b_{nk}v_{n} [/tex]



    -------------------------------------------------------------------


    [tex] h(L(v_{j}),v_{k}) = h(v_{j},L^{*}(v_{k})) [/tex]

    [tex] h( a_{1j}v_{1} +....+a_{nj}v_{n} , v_{k}) = h(v_{j}, b_{1j}v_{1}+....+b_{nk}v_{n}) [/tex]

    ................

    applying properties of "h"scalar product and properties of h-ohrtonormal bases

    ...............
    [tex] a_{jk}*h(v_{k},v_{k}) = \bar{b}_{kj}*h(v_{j},v_{j}) [/tex]

    [tex] a_{jk} = \bar{b}_{kj} [/tex]


    [tex] {b}_{jk} = \bar{a}_{kj} [/tex] !
     
  6. Sep 6, 2008 #5


    There are some typing errors, I've written "j" instead of "k" in some parts, the proof is ok
     
  7. Sep 6, 2008 #6

    Dick

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    Looks like the idea is right. Good job.
     
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