Results 1 to 13 of 13

Math Help - Homomorphism from the complex to 2*2 matrices

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    122

    Homomorphism from the complex to 2*2 matrices

    is there a homomorphism
    Φ:C×C→M₂(R)
    Φ must be onto
    I have tried variations on the type of entries in the 2 by 2 matrices that will preserve the additive and multiplicative properties of a homomorphism but this seems quite long. Is there a better way for me to find if a homomorphism exists that is onto?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,057
    Thanks
    1690
    Are you sure the question is whether there is a homomorphism between C\times C and M_2(R) and not just between C and M_2(R)?

    There is an obvious homomorphism a+ bi\rightarrow \begin{pmatrix}a & -b \\ b & a}\end{pmatrix}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2010
    Posts
    122
    yes i know that homomorphism it is in our text book.The question specifically asks for two input values and this homomorphism doesnt seem to preserve the multiplicative structure.I have tried a similar matrix.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by HallsofIvy View Post
    Are you sure the question is whether there is a homomorphism between C\times C and M_2(R) and not just between C and M_2(R)?

    There is an obvious homomorphism a+ bi\rightarrow \begin{pmatrix}a & -b \\ b & a}\end{pmatrix}.
    This is not a surjective homomorphism, however!
    If the question was just to find a homomorphism, there would always be the zero homomorphism.

    Think of \mathbb{C}^2 as a 4-dimensional vector space over \mathbb{R}. As a vector space, M_2(\mathbb{R}) is just the same thing!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2010
    Posts
    122
    i still cant figure this out. I see that there are 4 entries of real numbers but how can i use this fact to get a surgective homomorphism?
    the only thing i can think of is that 4 linearly independant 2*2 matrices will generate "all" real 2*2 matrices. But i dont think the the multiplication property of an isomorphism holds in the case where i take the matrix entries to be a,b,c,d.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by ulysses123 View Post
    i still cant figure this out. I see that there are 4 entries of real numbers but how can i use this fact to get a surgective homomorphism?
    the only thing i can think of is that 4 linearly independant 2*2 matrices will generate "all" real 2*2 matrices. But i dont think the the multiplication property of an isomorphism holds in the case where i take the matrix entries to be a,b,c,d.
    I think you are confused about the group operation! The set of square 2x2 matrices is not a group under matrix multiplication, because not all matrices are invertible. It is a group under addition though (just like \mathbb{C}^2).
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by Bruno J. View Post
    Think of \mathbb{C}^2 as a 4-dimensional vector space over \mathbb{R}. As a vector space, M_2(\mathbb{R}) is just the same thing!
    The problem with this is that they're isomorphic as vector spaces but what is asked is an onto ring homomorphism which actually doesn't exist since \mathbb{C}^2 is a commutative ring and M_2(\mathbb{R}) is not.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Mar 2010
    Posts
    122
    but does the homomorphism have to be comutative?
    When i chech the multiplication property F{(x,y)(u,v)}=F(x,y)F(u,v)
    Do i multiply components only or everything?

    eg
    F{(a+bi,c+di)(e+fi,g+hi)}
    doi have
    F{(ae-fi+afi+bei,cg-dh+chi+dgi)}
    or do i multiply everything?
    Because in this case a homorphism does exist even though it is not surjective, otherwise no homomorphism exists.
    Last edited by ulysses123; August 8th 2010 at 07:04 PM.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    General result I think is:

    There is homomorphism between \mathbb{R}^n and the field of matrices in M_n(\mathbb{R})

    (I think we proved this theorem on linear algebra 1)
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by ulysses123 View Post
    but does the homomorphism have to be comutative?
    When i chech the multiplication property F{(x,y)(u,v)}=F(x,y)F(u,v)
    Do i multiply components only or everything?

    eg
    F{(a+bi,c+di)(e+fi,g+hi)}
    doi have
    F{(ae-fi+afi+bei,cg-dh+chi+dgi)}
    or do i multiply everything?
    Because in this case a homorphism does exist even though it is not surjective, otherwise no homomorphism exists.
    In general the homomorphic image of a commutative ring (obviously a ring hom.) is commutative: Take f:A\rightarrow B where A is commutative and B is simply a ring, then without loss of generality f is onto (because f(A) is a subring) then take x,y\in B then x=f(a), \ y=f(b) for some a,b\in A then xy=f(a)f(b)=f(ab)=f(ba)=f(b)f(a)=yx.

    I'm assuming of course that you're giving \mathbb{C}^2 the usual structure of a product ring (or some commutative structure at least)
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by Jose27 View Post
    The problem with this is that they're isomorphic as vector spaces but what is asked is an onto ring homomorphism which actually doesn't exist since \mathbb{C}^2 is a commutative ring and M_2(\mathbb{R}) is not.
    That's true! I had read the problem too fast.
    Ulysses, are you looking for a group homomorphism, or a ring homomorphism?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by Also sprach Zarathustra View Post
    General result I think is:

    There is homomorphism between \mathbb{R}^n and the field of matrices in M_n(\mathbb{R})

    (I think we proved this theorem on linear algebra 1)
    The matrices M_n(\mathbb{R}) are not a field, no more than \mathbb{R}^n!

    In any case, even as vector spaces over \mathbb{R}, these two objects are non-isomorphic (not the same dimension).
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Member
    Joined
    Mar 2010
    Posts
    122
    the question specifcally says:is there ANY homomrphism, so i'm presuming he wants us to think about this question.
    I have basically showed 1 example that is a ring homomorphism but is not surjective, and a map that is surjective but does not preserve the ring homomorphism properties.And additionally stated that C*C is commutative but matrices aren't.
    Does a homomrphism exist though that is NOT necassarily a ring homomorphism?Because i have answered this as no, because i cant find any homomorphism that is surjective.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Invertible Complex Matrices
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: November 12th 2010, 07:39 PM
  2. Orders, Matrices, Complex entries...
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 24th 2010, 05:34 AM
  3. Matrices and complex numbers
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 22nd 2010, 03:29 AM
  4. Matrices / Complex Numbers /eigenvalues+vectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 29th 2010, 02:25 PM
  5. Complex Numbers- Matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 29th 2009, 10:27 AM

Search Tags


/mathhelpforum @mathhelpforum