A singular matrix can still be diagonalizable. Take [ 1 0 ] [ 0 0 ]
for example. So it wouldn't make sense to check if a matrix is singular. The best test is to compute the condition number of V as per my previous email. Patrick On 1/19/21 3:30 PM, Patrick Dupre wrote: > Thanks, but I think that gsl_eigen_nonsymmv must return an error or a warming > when dealing with singular matrices. > Of course, I just took a 3 x 3 matrix as an example, but I need to deal with > any size of matrices, and some maybe singular. > > Furthermore, maxima detects the singular matrices. Why not GSL? > Currently, we can "diagonalize" singular matrices, and deal with eigenvectors > as they were relevant eigenvectors. > > Indeed, gsl_eigen_nonsymmv could return at least an error if the matrix > determinant is zero > or if eigenvalues are degenerated. > I guess that the first thing that gsl_eigen_nonsymmv does is to calculate the > determinant. > >> Does this help? >> >> https://lists.gnu.org/archive/html/help-gsl/2005-02/msg00012.html >> >> If you get an error or a zero, the matrix cannot be inverted. >> >> Maybe you are asking "Isn't there a simpler way?" and I think the answer >> is "No." I can look at your matrix and see that it does not have full >> rank because columns 0 and 1 are perfectly correlated. But the condition >> that must apply to have full rank is more complex: No linear combination >> of columns can be perfectly correlated with any other linear >> combination. That's not hard to test with a 3x3 matrix, but it becomes >> impractical with a larger matrix. >> >> You could compute correlations between all pairs of columns and if any >> correlation is 1.0 then the matrix does not have full rank (and if it's >> very close to 1.0, then you will have stability issues). But there are >> matricies that will pass that test that do not have full rank. So that >> test is probably somewhat cheaper computationally, but has a known false >> negative rate and is not a good test. >> >> -Alan >> >> >> On 1/19/2021 4:05 PM, Patrick Dupre wrote: >>> This is perfectly correct. >>> >>> My question is how do I detect singular matrix since >>> gsl_eigen_nonsymmv >>> does not do it? >>> >>> >>>> Your matrix is singular (the first two columns are perfectly correlated, >>>> so your matrix does not have full rank). >>>> >>>> The standard test of singularity is that the determinant cannot be >>>> computed (you try to calculate it, and you get zero or a numerical >>>> error). This post describes the problem and a shortcut: >>>> >>>> https://stackoverflow.com/questions/13145948/how-to-find-out-if-a-matrix-is-singular >>>> >>>> -Alan >>>> >>>> >>>> On 1/19/2021 2:51 PM, Patrick Dupre wrote: >>>>> gsl_eigen_nonsymmv_workspace >>>>> has no member n_evals >>>>> >>>>> issue: >>>>> >>>>> Diagonalizing >>>>> double data_3 [] = { 0.0, 0.0, 1.0, >>>>> 0.0, 0.0, 0.0, >>>>> 0.0, 0.0, 0.0 } ; >>>>> >>>>> I get >>>>> eigenvalue = 0 +0i >>>>> eigenvector = >>>>> 1 +0i >>>>> 0 +0i >>>>> 0 +0i >>>>> eigenvalue = 0 +0i >>>>> eigenvector = >>>>> 0 +0i >>>>> 1 +0i >>>>> 0 +0i >>>>> eigenvalue = 0 +0i >>>>> eigenvector = >>>>> -1 +0i >>>>> 0 +0i >>>>> 3.00625e-292 +0i >>>>> >>>>> >>>>> which is wrong. >>>>> The last eigenvector is not correct because this matrix is not >>>>> diagonalizable. >>>>> >>>>> I need to identify such matrices. >>>>> >>>>> >>>>> =========================================================================== >>>>> Patrick DUPRÉ | | email: pdu...@gmx.com >>>>> Laboratoire interdisciplinaire Carnot de Bourgogne >>>>> 9 Avenue Alain Savary, BP 47870, 21078 DIJON Cedex FRANCE >>>>> Tel: +33 (0)380395988 >>>>> =========================================================================== >>>>> >>>>> >>>>>> Sent: Tuesday, January 19, 2021 at 6:56 PM >>>>>> From: "Patrick Alken" <patrick.al...@colorado.edu> >>>>>> To: help-gsl@gnu.org >>>>>> Subject: Re: eigensystem >>>>>> >>>>>> What do you mean by handle it? According to the documentation, if the >>>>>> function cannot compute all eigenvalues, an error code is returned. In >>>>>> the case of gsl_eigen_nonsymm, the number of converged eigenvalues is >>>>>> stored in w->n_evals. >>>>>> >>>>>> Patrick >>>>>> >>>>>> On 1/19/21 10:33 AM, Patrick Dupre wrote: >>>>>>> Hello, >>>>>>> >>>>>>> Is there a way to handle the possible error of gsl_eigen_nonsymmv ? >>>>>>> >>>>>>> For example, when the matrix is not diagonalizable. >>>>>>> >>>>>>> Thanks >>>>>>> >>>>>>> =========================================================================== >>>>>>> Patrick DUPRÉ | | email: pdu...@gmx.com >>>>>>> Laboratoire interdisciplinaire Carnot de Bourgogne >>>>>>> 9 Avenue Alain Savary, BP 47870, 21078 DIJON Cedex FRANCE >>>>>>> Tel: +33 (0)380395988 >>>>>>> =========================================================================== >>>>>>> >>>>>>> >>>> -- >>>> >>>> Alan D. Mead, Ph.D. >>>> President, Talent Algorithms Inc. >>>> >>>> science + technology = better workers >>>> >>>> http://secure-web.cisco.com/1zrHBOw9QrpIw_ie7c5QyyUd_QUL-cVnNWKqQN9USFmbOp-vM-l5Uqsy_dFAeR6f9g1YHWdlRNeB-nq7X7q2ZdNslNHy4Cb-JvNu8PSh0Wi4S5w9pGr2E1J0JvrwmtEHCU7-KZ93v17SsUX4UliTxQ3B-2iL-nNYDy8nMHy3J9yw5mxCElCLDrFy6nYwe2a5Q6OFNISlShBpHF86jGJcpuIdwunE_c90vAl6p4qAsTCQcBgtyYdhwUANjkKDHLoqB5aI2Vhg8sdtmh4ezPeDevhBwMZpARDj54O1dsN10IN9oIweqA_gU25vfuPRB-y-FiaU4mnaLkguIbsMpe_0NfgYCSxAluQRizwiRYy59aOaxAnTsJI8dNdjksEmJXrwGgYX8FZGfxnL70A2WDL3X55BTJe7dAAexZmqZrqBU_-ZWPdPOKilv4IGBVsFrAaEsv1iX7ISkYW1ORQdJyla-5Q/http%3A%2F%2Fwww.alanmead.org >>>> >>>> The irony of this ... is that the Internet is >>>> both almost-infinitely expandable, while at the >>>> same time constrained within its own pre-defined >>>> box. And if that makes no sense to you, just >>>> reflect on the existence of Facebook. We have >>>> the vastness of the internet and yet billions >>>> of people decided to spend most of them time >>>> within a horribly designed, fake-news emporium >>>> of a website that sucks every possible piece of >>>> personal information out of you so it can sell it >>>> to others. And they see nothing wrong with that. >>>> >>>> -- Kieren McCarthy, commenting on why we are not >>>> all using IPv6 >>>> >>>> >> -- >> >> Alan D. Mead, Ph.D. >> President, Talent Algorithms Inc. >> >> science + technology = better workers >> >> http://secure-web.cisco.com/1zrHBOw9QrpIw_ie7c5QyyUd_QUL-cVnNWKqQN9USFmbOp-vM-l5Uqsy_dFAeR6f9g1YHWdlRNeB-nq7X7q2ZdNslNHy4Cb-JvNu8PSh0Wi4S5w9pGr2E1J0JvrwmtEHCU7-KZ93v17SsUX4UliTxQ3B-2iL-nNYDy8nMHy3J9yw5mxCElCLDrFy6nYwe2a5Q6OFNISlShBpHF86jGJcpuIdwunE_c90vAl6p4qAsTCQcBgtyYdhwUANjkKDHLoqB5aI2Vhg8sdtmh4ezPeDevhBwMZpARDj54O1dsN10IN9oIweqA_gU25vfuPRB-y-FiaU4mnaLkguIbsMpe_0NfgYCSxAluQRizwiRYy59aOaxAnTsJI8dNdjksEmJXrwGgYX8FZGfxnL70A2WDL3X55BTJe7dAAexZmqZrqBU_-ZWPdPOKilv4IGBVsFrAaEsv1iX7ISkYW1ORQdJyla-5Q/http%3A%2F%2Fwww.alanmead.org >> >> The irony of this ... is that the Internet is >> both almost-infinitely expandable, while at the >> same time constrained within its own pre-defined >> box. And if that makes no sense to you, just >> reflect on the existence of Facebook. We have >> the vastness of the internet and yet billions >> of people decided to spend most of them time >> within a horribly designed, fake-news emporium >> of a website that sucks every possible piece of >> personal information out of you so it can sell it >> to others. And they see nothing wrong with that. >> >> -- Kieren McCarthy, commenting on why we are not >> all using IPv6 >> >> >
