Results 1 to 15 of 15

Math Help - Same Eigenvalues

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    41

    Question Same Eigenvalues

    Quantum computation and quantum ... - Google Books

    I am doing the problem above.

    A_{ij} = < v_i| T | v_j> then use the fact that I = \sum_m |w_m><w_m| and I = \sum_n |w_n><w_n|

    eventually you get


    A_{ij} = \sum_m \sum_n <v_i|w_m><w_n|v_j>B_{mn}

    how do I invoke the unitary matrix? We know that U=|w_i><v_i|
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    I assume you're doing Exercise 2.20? What is B_{mn}?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2009
    Posts
    41
    It's the same as in the question (B_{ij}) except your introducing new indices m and n because I already used i and j. Is that clear?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Is that clear?
    Not exactly. Are you doing Exercise 2.20?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2009
    Posts
    41
    Yes. Sorry.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    I don't see a matrix B anywhere in the problem statement.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    What does |\rangle mean?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Mar 2009
    Posts
    41
    In this case B would be their A_{ij}^{''}


    Is this the appropriate forum I should have posted in?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Drexel28: in quantum mechanics, "kets", or column vectors (at least in finite-dimensional Hilbert spaces) look like this: |v\rangle, whereas "bras", or row vectors in the dual space look like this: \langle w|. A "bracket" is an inner product of a bra with a ket: \langle w|v\rangle. You can also form a matrix with the outer product thus:

    \displaystyle\sum_{i}|w_{i}\rangle\langle v_{i}|.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Nusc:

    This forum is just fine. You could have gone either here or in the Advanced Applied Math forum. The underlying math is definitely linear algebra, though, so you're fine.

    So you're claiming that

    \displaystyle A_{ij}'=\sum_{m}\sum_{n}\langle v_{i}|w_{m}\rangle\langle w_{n}|v_{j}\rangle A_{ij}''?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Ackbeet View Post
    Drexel28: in quantum mechanics, "kets", or column vectors (at least in finite-dimensional Hilbert spaces) look like this: |v\rangle, whereas "bras", or row vectors in the dual space look like this: \langle w|. A "bracket" is an inner product of a bra with a ket: \langle w|v\rangle. You can also form a matrix with the outer product thus:

    \displaystyle\sum_{i}|w_{i}\rangle\langle v_{i}|.
    Last question (there's a chance I can help if I understand this) if one has some finite dimensional Hilbert space \mathcal{H} with inner product \langle\cdot,\cdot\rangle and \varphi\in\text{Hom}\left(\mathcal{H},F\right) (a covector if that notation is unfamiliar...I know the terminology covector is common in physics, right?) what does \langle \varphi,v\rangle even mean for v\in\mathcal{H}?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Junior Member
    Joined
    Mar 2009
    Posts
    41
    That's correct. Then somehow I am supposed to come up with U_{im} and U_{nj}^{*} but I thought it was the other way around U_{mn} and U_{ij}^{*}.

    Then I am supposed to show that they have the same eigenvalues. Similar matrices have the same eigenvalues but I am not sure how relevant that is based on what is given.

    Answer is:
     A_{ij}'= U_{im} U_{nj}^{*} A_{ij}'' which is unclear.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Reply to Drexel28: You're way over my head there. I do know this: a Hilbert space is a complete inner product space, by definition. So with any Hilbert space, there is an inner product, and it's defined for all the vectors in the Hilbert space.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Reply to Nusc:

    Why not try this:

    A_{ij}'=\langle v_{i}|A|v_{j}\rangle implies

    \displaystyle\sum_{i}|v_{i}\rangle A_{ij}'=\sum_{i}|v_{i}\rangle\langle v_{i}|A|v_{j}\rangle=A|v_{j}\rangle, and hence

    \displaystyle\sum_{i,j}|v_{i}\rangle A_{ij}'\langle v_{j}|=\sum_{j}A|v_{j}\rangle\langle v_{j}|=A.

    Similarly,

    \displaystyle\sum_{i,j}|w_{i}\rangle A_{ij}''\langle w_{j}|=A.

    Therefore,

    \displaystyle\sum_{i,j}|w_{i}\rangle A_{ij}''\langle w_{j}|<br />
=\sum_{i,j}|v_{i}\rangle A_{ij}'\langle v_{j}|.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Junior Member
    Joined
    Mar 2009
    Posts
    41
    That's much more elegant!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 27th 2010, 01:08 PM
  2. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 6th 2009, 06:27 AM
  3. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 7th 2008, 03:08 AM
  4. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 3rd 2008, 10:35 AM
  5. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 30th 2008, 07:11 PM

Search Tags


/mathhelpforum @mathhelpforum