Results 1 to 8 of 8

Math Help - theorem about commuting matrices.

  1. #1
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    theorem about commuting matrices.

    Hi

    I was reading a book on quantum mechanics (thats branch of physics). There is this
    theorem which I didn't understand. Now the theorem is about physical observable
    quantities like energy,momentum etc.. In quantum mechanics , such physical
    observable quantity is represented by an operator. In matrix mechanics , which is a
    version of quantum mechanics , such operators are written mathematically as
    matrices. So the theorem is basically about matrices. I am going to translate the
    theorem in mathematical language , removing words which relate to physics.

    Theorem-- If two matrices commute, they possess a common set of eigenvectors.
    This is true for both degenerate and non-degenerate eigenvectors.

    Now when we have non-degenerate eigenvectors , two such commuting matrices
    happen to have common eigenvectors anyway. But I have a question about
    degenerate case. In such case , how do we obtain the common eigenvectors ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2

    Re: theorem about commuting matrices.

    Quote Originally Posted by issacnewton View Post
    two such commuting matrices happen to have common eigenvectors anyway.
    This may be the important point. Find a copy of a proof and see if it uses the words "if and only if". If so, determine how it is proven in both directions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: theorem about commuting matrices.

    physics books are sloppy when it comes to math "proofs". I don't understand what the author is talking. So I wanted to ask people here. math people are better educated in math than physicists.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Re: theorem about commuting matrices.

    Quote Originally Posted by issacnewton View Post
    Hi

    I was reading a book on quantum mechanics (thats branch of physics). There is this
    theorem which I didn't understand. Now the theorem is about physical observable
    quantities like energy,momentum etc.. In quantum mechanics , such physical
    observable quantity is represented by an operator. In matrix mechanics , which is a
    version of quantum mechanics , such operators are written mathematically as
    matrices. So the theorem is basically about matrices. I am going to translate the
    theorem in mathematical language , removing words which relate to physics.

    Theorem-- If two matrices commute, they possess a common set of eigenvectors.
    This is true for both degenerate and non-degenerate eigenvectors.

    Now when we have non-degenerate eigenvectors , two such commuting matrices
    happen to have common eigenvectors anyway. But I have a question about
    degenerate case. In such case , how do we obtain the common eigenvectors ?
    we also need the base field k to be algebraically closed. so we have a finite dimensional k-vector space V and the operators T_1,T_2 : V \longrightarrow V with T_1T_2=T_2T_1. since k is algebraically closed, T_2(v)=\lambda v for some \lambda \in k and 0 \neq v \in V. let

    W=\{x \in V: \ T_2(x)=\lambda x \}.

    then W is a non-zero subspace of V and for any x \in W we have T_2T_1(x)=T_1T_2(x)=\lambda T_1(x). so T_1(x) \in W. thus T':=T_1|_W is an operator on W. let u \in W be any eigenvector of T'. clearly u is also an eigenvector of T_2 because u \in W. hence u is an eigenvector of both T_1 and T_2.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2

    Re: theorem about commuting matrices.

    Quote Originally Posted by issacnewton View Post
    Hi

    I was reading a book on quantum mechanics (thats branch of physics). There is this
    theorem which I didn't understand. Now the theorem is about physical observable
    quantities like energy,momentum etc.. In quantum mechanics , such physical
    observable quantity is represented by an operator. In matrix mechanics , which is a
    version of quantum mechanics , such operators are written mathematically as
    matrices. So the theorem is basically about matrices. I am going to translate the
    theorem in mathematical language , removing words which relate to physics.

    Theorem-- If two matrices commute, they possess a common set of eigenvectors.
    This is true for both degenerate and non-degenerate eigenvectors.

    Now when we have non-degenerate eigenvectors , two such commuting matrices
    happen to have common eigenvectors anyway. But I have a question about
    degenerate case. In such case , how do we obtain the common eigenvectors ?
    Observables in quantum mechanics are always represented by Hermitian operators. The eigenvectors of an Hermitian operator can be chosen to form an orthonormal basis, which implies that they are not degenerate. At least, while you may have degenerate eigenvalues, the geometric multiplicity of degenerate eigenvalues of an Hermitian operator is equal to its algebraic multiplicity. So I'm not entirely sure why you're talking about degenerate eigenvectors. In the degenerate eigenvalue case, you find the eigenvectors in the usual manner. You're not even talking about generalized eigenvectors here.

    Does that answer your question?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: theorem about commuting matrices.

    Noncomm, thanks for the proof. makes thing clear.

    ackbeet , oh yes , i am actually talking about the degenerate eigenvalues ,
    not the degenerate eigenvectors..

    after checking with the problem i was having , i am able to get the set of
    common eigenvectors for the two commuting hermitian matrices having degenerate
    eigenvalues. i needed to choose the indeterminate constants properly so that
    the set of eigenvectors matched.

    but the theorem i quoted doesn't say anything about the hermitian nature of the
    matrices. so is it more general theorem , valid for non hermitian commuting matrices as well ?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2

    Re: theorem about commuting matrices.

    Quote Originally Posted by issacnewton View Post
    Ackbeet , oh yes , i am actually talking about the degenerate eigenvalues ,
    not the degenerate eigenvectors..

    after checking with the problem i was having , i am able to get the set of
    common eigenvectors for the two commuting hermitian matrices having degenerate
    eigenvalues. i needed to choose the indeterminate constants properly so that
    the set of eigenvectors matched.

    but the theorem i quoted doesn't say anything about the hermitian nature of the
    matrices. so is it more general theorem , valid for non hermitian commuting matrices as well ?
    According to this wiki, two matrices commute if and only if they are simultaneously diagonalizable. So it appears to be more general than just Hermitian matrices.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: theorem about commuting matrices.

    thanks for pointing in the right direction......

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Commuting p-cycles in Sn
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: October 16th 2011, 01:50 AM
  2. Singular Matrices Theorem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 16th 2010, 02:53 AM
  3. Commuting Matrices
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: August 20th 2010, 08:15 AM
  4. Spectral radius of sum of commuting matrices
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: July 3rd 2010, 08:47 PM
  5. Commuting Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 5th 2009, 02:29 PM

/mathhelpforum @mathhelpforum