Comments on: Simultaneous Diagonalization of Hermitian Matrices
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/
Quantum Mechanics problems and solutions by Philippine science studentsTue, 21 May 2019 12:19:54 +0000hourly1https://wordpress.org/?v=4.9.10By: Muhammad Nadeem
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-26028
Tue, 13 Oct 2015 18:27:56 +0000http://www.quantumsciencephilippines.com/?p=216#comment-26028Thank you so much for this. Its very nice explanation for me. good work
]]>By: Jram
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-6375
Sun, 09 Mar 2014 05:01:08 +0000http://www.quantumsciencephilippines.com/?p=216#comment-6375Here, how did you choose the unitary matrix???
Or is it that the matrix you chosen is the eigen vectors written one after the other..?? What is the mathematics behind it??
]]>By: Daniel Huber
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-3748
Thu, 14 Nov 2013 12:57:13 +0000http://www.quantumsciencephilippines.com/?p=216#comment-3748Sorry,I was a bit too fast and not accurate enough. Here is a more accurate description:
If you want an unitary transformation matrix for diagonalization, the matrizes must commute and they then have a common basis from eigenvectors.
For non-commuting matrizes, you must solve the generalized eigenvalue problem and you get in general a non unitary transformation matrix.
cheers, Daniel
]]>By: Daniel Huber
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-3747
Thu, 14 Nov 2013 11:53:58 +0000http://www.quantumsciencephilippines.com/?p=216#comment-3747Hi MARYJANE,
thank’s for your effort, but you picked a special case that is not generally valid.
First,as other peole before me already pointed out, you need to mention that both matrizes must commute to be simultaneously diagonazable.
But then, the matrix that does this, can not be obtained from the two seperate eigenvalue problems, but you must solve the generalized eigen value problem: A x= lamda B x
where A and B are the 2 given matrizes, lambda the eigenvalue and x the eigenvector. The colum matrix of the eigenvectors: T will then diagonalize both A and B: T A Transpose[T] and T B Transpose[T]
]]>By: John
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-2008
Mon, 11 Feb 2013 08:33:18 +0000http://www.quantumsciencephilippines.com/?p=216#comment-2008Very nice.Many thanks.
]]>By: Luigi
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-951
Sun, 13 Mar 2011 17:39:00 +0000http://www.quantumsciencephilippines.com/?p=216#comment-951You should stres that you can do it only because the matrices that you have choosen commute…
]]>By: shahab
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-938
Tue, 27 Apr 2010 16:09:51 +0000http://www.quantumsciencephilippines.com/?p=216#comment-938I see…
thanks.
]]>By: Kabam_101
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-916
Wed, 07 Apr 2010 23:12:15 +0000http://www.quantumsciencephilippines.com/?p=216#comment-916@shahab: I don’t see any trick on this. Everything is just fine. The eigenvectors were not just “chosen” but actually solved from the other matrix (\Lambda) which commutes with the matrix (\Omega) you wanted to diagonalize. The only important thing missing here is that it wasn’t mentioned that this method only works if both matrices commute.
]]>By: shahab
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-915
Mon, 05 Apr 2010 22:53:16 +0000http://www.quantumsciencephilippines.com/?p=216#comment-915well im not too sure but i think you have tricked us here….
in practice one doesnt happen to choose the values for the e’vectors of the degenerate matrix that fits the other one…!
]]>By: Chen Feng
https://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/comment-page-1/#comment-718
Thu, 10 Dec 2009 20:54:15 +0000http://www.quantumsciencephilippines.com/?p=216#comment-718Hello, MARYJANE. I just want to point out that the simultaneous denationalization works here because your matrices commute (\Omega \Gamma = \Gamma \Omega). In general, we can’t simultaneously diagonalize two matrices. Please refer to page 235 of the book ‘Matrix Analysis’ by Horn R A. And email me if you want to discuss more about this important research issue.
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