Dear Python-users, I am currently very confused about the Scipy routine to obtain the eigenvectors of a complex matrix.In attached you find two files to diagonalize a 2X2 complex Hermitian matrix, however, on my computer, If I run python, I got: [[ 0.80322132+0.j 0.59500941+0.02827207j] [-0.59500941+0.02827207j 0.80322132+0.j ]] If I compile the fortran code, I got: ( -0.595009410289, -0.028272068905) ( 0.802316135182, 0.038122316497) ( -0.803221321796, 0.000000000000) ( -0.595680709955, 0.000000000000)
From the scipy webpage, it is said that numpy.linalg.eig() provides nothing butan interface to lapack zheevd subroutine, which is used in my fortran code. Would somebody be kind to tell me how to get consistent results? Many thanks in advance. Best wishes, Hongbin
Ad hoc, ad loc and quid pro quo --- Jeremy Hilary Boob
2012/4/2 Hongbin Zhang <hongbin_zhang82@hotmail.com>:
Dear Python-users,
I am currently very confused about the Scipy routine to obtain the eigenvectors of a complex matrix. In attached you find two files to diagonalize a 2X2 complex Hermitian matrix, however, on my computer,
If I run python, I got:
[[ 0.80322132+0.j 0.59500941+0.02827207j] [-0.59500941+0.02827207j 0.80322132+0.j ]]
If I compile the fortran code, I got:
( -0.595009410289, -0.028272068905) ( 0.802316135182, 0.038122316497) ( -0.803221321796, 0.000000000000) ( -0.595680709955, 0.000000000000)
these results look more like eigh (except flipped)
numpy.linalg.eigh(numpy.array(H))[1] array([[ 0.59568071+0.j , -0.80322132+0.j ], [ 0.80231613-0.03812232j, 0.59500941-0.02827207j]])
Josef
From the scipy webpage, it is said that numpy.linalg.eig() provides nothing but an interface to lapack zheevd subroutine, which is used in my fortran code.
< /div> Would somebody be kind to tell me how to get consistent results?
Many thanks in advance.
Best wishes,
Hongbin
Ad hoc, ad loc and quid pro quo
--- Jeremy Hilary Boob
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Both results are correct. There are 2 factors that make the results look different: 1) The order: the 2nd eigenvector of the numpy solution corresponds to the 1st eigenvector of your solution, note that the vectors are written in columns. 2) The phase: an eigenvector can be multiplied by an arbitrary phase factor with absolute value = 1. As you can see this factor is -1 for the 2nd eigenvector and -0.99887305445887753-0.047461785427773337j for the other one. Val 2012/4/2 Hongbin Zhang <hongbin_zhang82@hotmail.com>
Dear Python-users,
I am currently very confused about the Scipy routine to obtain the eigenvectors of a complex matrix. In attached you find two files to diagonalize a 2X2 complex Hermitian matrix, however, on my computer,
If I run python, I got:
[[ 0.80322132+0.j 0.59500941+0.02827207j] [-0.59500941+0.02827207j 0.80322132+0.j ]]
If I compile the fortran code, I got:
( -0.595009410289, -0.028272068905) ( 0.802316135182, 0.038122316497) ( -0.803221321796, 0.000000000000) ( -0.595680709955, 0.000000000000)
From the scipy webpage, it is said that numpy.linalg.eig() provides nothing but an interface to lapack zheevd subroutine, which is used in my fortran code.
< /div> Would somebody be kind to tell me how to get consistent results?
Many thanks in advance.
Best wishes,
Hongbin
Ad hoc, ad loc and quid pro quo
--- Jeremy Hilary Boob
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Hi, On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky@gmail.com> wrote:
Both results are correct. There are 2 factors that make the results look different: 1) The order: the 2nd eigenvector of the numpy solution corresponds to the 1st eigenvector of your solution, note that the vectors are written in columns. 2) The phase: an eigenvector can be multiplied by an arbitrary phase factor with absolute value = 1. As you can see this factor is -1 for the 2nd eigenvector and -0.99887305445887753-0.047461785427773337j for the other one.
Thanks for this answer; for my own benefit: Definition: A . v = L . v where A is the input matrix, L is an eigenvalue of A and v is an eigenvector of A. http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix In [63]: A = [[0.6+0.0j, -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]] In [64]: L, v = np.linalg.eig(A) In [66]: np.allclose(np.dot(A, v), L * v) Out[66]: True Best, Matthew
BTW this extra degree of freedom can be used to "rotate" the eigenvectors along the unit circle (multiplication by exp(j*phi)). To those of physical inclinations it should remind of gauge fixing (vector potential in EM/QM). These "rotations" can be used to make one (any) non-zero component of each eigenvector be positive real number. Finally to the point: it seems that numpy.linalg.eig uses these "rotations" to turn the diagonal elements in the eigenvector matrix to real positive numbers, that's why the numpy solutions looks neat. Val PS Probably nobody cares to know, but the phase factor I gave in my 1st email should be negated: 0.99887305445887753+0.047461785427773337j On Mon, Apr 2, 2012 at 8:53 PM, Matthew Brett <matthew.brett@gmail.com>wrote:
Hi,
Both results are correct. There are 2 factors that make the results look different: 1) The order: the 2nd eigenvector of the numpy solution corresponds to
On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky@gmail.com> wrote: the
1st eigenvector of your solution, note that the vectors are written in columns. 2) The phase: an eigenvector can be multiplied by an arbitrary phase factor with absolute value = 1. As you can see this factor is -1 for the 2nd eigenvector and -0.99887305445887753-0.047461785427773337j for the other one.
Thanks for this answer; for my own benefit:
Definition: A . v = L . v where A is the input matrix, L is an eigenvalue of A and v is an eigenvector of A.
http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix
In [63]: A = [[0.6+0.0j, -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]]
In [64]: L, v = np.linalg.eig(A)
In [66]: np.allclose(np.dot(A, v), L * v) Out[66]: True
Best,
Matthew _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Hej Val, Thank you very much for your replies. Yes, I know that both eigenvectors are correct while they are indeed related to each other by unitary transformations (unitary matrices). Actually, what I am trying to do is to evaluate the Berry phase which is closely related to the gauge chosen. It is okay to apply an arbitrary phase to the eigenvectors, while to get the (meaningful) physical quantity the phase should be consistent for all the other eigenvectors. To my understanding, if I run both Fortran and python on the same computer, they should have the same phase (that is the arbitrary phase is computer-dependent). Maybe some additional "rotations" have been performed in python, but should this be written/commented somewhere in the man page? I will try to fix this by performing additional rotation to make the diagonal elements real and check whether this is the solution or not. Thank you all again, and of course more insightful suggestions are welcome. Regards, Hongbin Ad hoc, ad loc and quid pro quo --- Jeremy Hilary Boob Date: Mon, 2 Apr 2012 22:19:55 -0500 From: kalatsky@gmail.com To: numpy-discussion@scipy.org Subject: Re: [Numpy-discussion] One question about the numpy.linalg.eig() routine BTW this extra degree of freedom can be used to "rotate" the eigenvectors along the unit circle (multiplication by exp(j*phi)). To those of physical inclinations it should remind of gauge fixing (vector potential in EM/QM). These "rotations" can be used to make one (any) non-zero component of each eigenvector be positive real number. Finally to the point: it seems that numpy.linalg.eig uses these "rotations" to turn the diagonal elements in the eigenvector matrix to real positive numbers, that's why the numpy solutions looks neat. Val PS Probably nobody cares to know, but the phase factor I gave in my 1st email should be negated: 0.99887305445887753+0.047461785427773337j On Mon, Apr 2, 2012 at 8:53 PM, Matthew Brett <matthew.brett@gmail.com> wrote: Hi, On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky@gmail.com> wrote:
Both results are correct.
There are 2 factors that make the results look different:
1) The order: the 2nd eigenvector of the numpy solution corresponds to the
1st eigenvector of your solution,
note that the vectors are written in columns.
2) The phase: an eigenvector can be multiplied by an arbitrary phase factor
with absolute value = 1.
As you can see this factor is -1 for the 2nd eigenvector
and -0.99887305445887753-0.047461785427773337j for the other one.
Thanks for this answer; for my own benefit: Definition: A . v = L . v where A is the input matrix, L is an eigenvalue of A and v is an eigenvector of A. http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix In [63]: A = [[0.6+0.0j, -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]] In [64]: L, v = np.linalg.eig(A) In [66]: np.allclose(np.dot(A, v), L * v) Out[66]: True Best, Matthew _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Dears, Though it might sounds strange, but the eigenvectors of my 2X2 matrix is rather different if I get it calculated in a loop over many other similar matrices: for instance: matrix: [[ 0.60000000+0.j -1.97537668-0.09386068j] [-1.97537668+0.09386068j -0.60000000+0.j ]] eigenvals: [-2.06662112 2.06662112] eigenvects: [[ 0.59568071+0.j 0.80231613-0.03812232j] [-0.80322132+0.j 0.59500941-0.02827207j]] In this case, the elements in the first column of the eigenvectors are real. In the fortran code, such transformation can be easily done by dividing all the elements in the i-th row by EV_{i1}/abs(EV_{i1}, where EV_{i1} denotes the first element in the i-th row. Same can be performed column-wise if it is intended. In this way, at least for the moment, I could get the same eigenvectors for the same complex matrix by Python and Fortran. I do not know whether this is the solution, but I hope this would work. Cheers, Hongbin Ad hoc, ad loc and quid pro quo --- Jeremy Hilary Boob From: hongbin_zhang82@hotmail.com To: numpy-discussion@scipy.org Date: Tue, 3 Apr 2012 15:02:18 +0800 Subject: Re: [Numpy-discussion] One question about the numpy.linalg.eig() routine Hej Val, Thank you very much for your replies. Yes, I know that both eigenvectors are correct while they are indeed related to each other by unitary transformations (unitary matrices). Actually, what I am trying to do is to evaluate the Berry phase which is closely related to the gauge chosen. It is okay to apply an arbitrary phase to the eigenvectors, while to get the (meaningful) physical quantity the phase should be consistent for all the other eigenvectors. To my understanding, if I run both Fortran and python on the same computer, they should have the same phase (that is the arbitrary phase is computer-dependent). Maybe some additional "rotations" have been performed in python, but should this be written/commented somewhere in the man page? I will try to fix this by performing additional rotation to make the diagonal elements real and check whether this is the solution or not. Thank you all again, a nd of course more insightful suggestions are welcome. Regards, Hongbin Ad hoc, ad loc and quid pro quo &n bsp; --- Jeremy Hilary Boob Date: Mon, 2 Apr 2012 22:19:55 -0500 From: kalatsky@gmail.com To: numpy-discussion@scipy.org Subject: Re: [Numpy-discussion] One question about the numpy.linalg.eig() routine BTW this extra degree of freedom can be used to "rotate" the eigenvectors along the unit circle (multiplication by exp(j*phi)). To those of physical inclinations it should remind of gauge fixing (vector potential in EM/QM). These "rotations" can be used to make one (any) non-zero component of each eigenvector be positive real number. Finally to the point: it seems that numpy.linalg.eig uses these "rotations" to turn the diagonal elements in the eigenvector matrix to real positive numbers, that's why the numpy solutions looks neat. Val PS Probably nobody cares to know, but the phase factor I gave in my 1st email should be negated: 0.99887305445887753+0.047461785427773337j On Mon, Apr 2, 2012 at 8:53 PM, Matthew Brett <matthew.brett@gmail.com> wrote: Hi, On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky@gmail.com> wrote:
Both results are correct.
There are 2 factors that make the results look different:
1) The order: the 2nd eigenvector of the numpy solution corresponds to the
1st eigenvector of your solution,
note that the vectors are written in columns.
2) The phase: an eigenvector can be multiplied by an arbitrary phase factor
with absolute value = 1.
As you can see this factor is -1 for the 2nd eigenvector
and -0.99887305445887753-0.047461785427773337j for the other one.
Thanks for this answer; for my own benefit: Definition: A . v = L . v where A is the input matrix, L is an eigenvalue of A and v is an eigenvector of A. http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix In [63]: A = [[0.6+0.0j, -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]] In [64]: L, v = np.linalg.eig(A) In [66]: np.allclose(np.dot(A, v), L * v) Out[66]: True Best, Matthew _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Interesting. I happen to know a little bit about Berry's phase http://keck.ucsf.edu/~kalatsky/publications/PRL1998_BerryPhaseForLargeSpins.... http://keck.ucsf.edu/~kalatsky/publications/PRA1999_SpectraOfLargeSpins-Gene... The latter one knocks out all point groups. Probably you want to do something different, I cared about eigenvalues only (BTW my Hamiltonians were carefully crafted). Cheers Val PS I doubt anybody on this list cares to hear more about Berry's phase, should take this discussion off-line 2012/4/3 Hongbin Zhang <hongbin_zhang82@hotmail.com>
Hej Val,
Thank you very much for your replies.
Yes, I know that both eigenvectors are correct while they are indeed related to each other by unitary transformations (unitary matrices).
Actually, what I am trying to do is to evaluate the Berry phase which is closely related to the gauge chosen. It is okay to apply an arbitrary phase to the eigenvectors, while to get the (meaningful) physical quantity the phase should be consistent for all the other eigenvectors.
To my understanding, if I run both Fortran and python on the same computer, they should have the same phase (that is the arbitrary phase is computer-dependent). Maybe some additional "rotations" have been performed in python, but should this be written/commented somewhere in the man page?
I will try to fix this by performing additional rotation to make the diagonal elements real and check whether this is the solution or not.
Thank you all again, a nd of course more insightful suggestions are welcome.
Regards,
Hongbin
Ad hoc, ad loc and quid pro quo
&n bsp; --- Jeremy Hilary Boob
------------------------------ Date: Mon, 2 Apr 2012 22:19:55 -0500 From: kalatsky@gmail.com To: numpy-discussion@scipy.org Subject: Re: [Numpy-discussion] One question about the numpy.linalg.eig() routine
BTW this extra degree of freedom can be used to "rotate" the eigenvectors along the unit circle (multiplication by exp(j*phi)). To those of physical inclinations it should remind of gauge fixing (vector potential in EM/QM). These "rotations" can be used to make one (any) non-zero component of each eigenvector be positive real number. Finally to the point: it seems that numpy.linalg.eig uses these "rotations" to turn the diagonal elements in the eigenvector matrix to real positive numbers, that's why the numpy solutions looks neat. Val
PS Probably nobody cares to know, but the phase factor I gave in my 1st email should be negated: 0.99887305445887753+0.047461785427773337j
On Mon, Apr 2, 2012 at 8:53 PM, Matthew Brett <matthew.brett@gmail.com>wrote:
Hi,
Both results are correct. There are 2 factors that make the results look different: 1) The order: the 2nd eigenvector of the numpy solution corresponds to
On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky@gmail.com> wrote: the
1st eigenvector of your solution, note that the vectors are written in columns. 2) The phase: an eigenvector can be multiplied by an arbitrary phase factor with absolute value = 1. As you can see this factor is -1 for the 2nd eigenvector and -0.99887305445887753-0.047461785427773337j for the other one.
Thanks for this answer; for my own benefit:
Definition: A . v = L . v where A is the input matrix, L is an eigenvalue of A and v is an eigenvector of A.
http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix
In [63]: A = [[0.6+0.0j, -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]]
In [64]: L, v = np.linalg.eig(A)
In [66]: np.allclose(np.dot(A, v), L * v) Out[66]: True
Best,
Matthew _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Hi, 2012/4/2 Hongbin Zhang <hongbin_zhang82@hotmail.com>:
Dear Python-users,
I am currently very confused about the Scipy routine to obtain the eigenvectors of a complex matrix. In attached you find two files to diagonalize a 2X2 complex Hermitian matrix, however, on my computer,
If I run python, I got:
[[ 0.80322132+0.j 0.59500941+0.02827207j] [-0.59500941+0.02827207j 0.80322132+0.j ]]
If I compile the fortran code, I got:
( -0.595009410289, -0.028272068905) ( 0.802316135182, 0.038122316497) ( -0.803221321796, 0.000000000000) ( -0.595680709955, 0.000000000000)
From the scipy webpage, it is said that numpy.linalg.eig() provides nothing but an interface to lapack zheevd subroutine, which is used in my fortran code.
< /div> Would somebody be kind to tell me how to get consistent results?
I should also point out that matlab and octave give the same answer as your Fortran routine: octave:15> H=[0.6+0.0j, -1.97537668-0.09386068j; -1.97537668+0.09386068j, -0.6+0.0j] H = 0.60000 + 0.00000i -1.97538 - 0.09386i -1.97538 + 0.09386i -0.60000 + 0.00000i Best, Matthew
participants (4)
-
Hongbin Zhang -
josef.pktd＠gmail.com -
Matthew Brett -
Val Kalatsky