On Sat, Sep 10, 2016 at 2:18 PM, Yichao Yu <yyc1...@gmail.com> wrote:
> > > On Sat, Sep 10, 2016 at 2:12 PM, Stuart Brorson <s...@cloud9.net> wrote: > >> I don't think you can define that in a continuous way. >>> In general, if your application relies on certain property of the basis, >>> you should just normalize it that way. If you don't have a requirement >>> than >>> you should worry about it. >>> >> >> Thanks for the thoughts. >> >> I did a little more thinking and Googling. My conclusion: >> >> Yes, you can define the concept of orientation very clearly. The >> orientation of a vector space is given by the determinant of the >> matrix formed by its basis set (eigenvectors). The determinant will >> be either + or -1 for an orthonormal set of basis vectors. The sign >> of the determinant divides the orthonormal set into two equivalence >> classes of basis vectors, one with + orientation, the other with -. >> > > > What I actually mean is that you can constraint the result of eigs. > Obviously I meant "can't" . You can certainly flip signs to get the orientation you want but not in a meaningful and unique way. > >> >> I guess the concept of eigenvector/eigevalue is more general than the >> concept of orientation. Put another way, you can have a basis set >> (eigenvectors) without defining the orientation property, but you >> can't have orientation without first having a basis set. >> >> The OP's results from Matlab and Julia belong to opposite >> equivalence classes, which is what bothers me. However, >> from the standpoint of getting a basis, the concept of orientation is >> not required. One would need to impose some additional constraint on >> the eigenvalue decomposition algorithm in order to always get an >> eigenvector set with the same orientation, and I guess that additional >> constraint isn't important for the problems usually solved by >> eigendecomposition. >> >> Thanks for making me think about this, >> >> Stuart >> >> >> >> >> On Sat, 10 Sep 2016, Yichao Yu wrote: >> >> On Sat, Sep 10, 2016 at 12:20 PM, Stuart Brorson <s...@cloud9.net> wrote: >>> >>> Just a question from a non-mathematician. Also, this is a math >>>> question, not a Julia/Matlab question. >>>> >>>> I agree that Matlab and Julia are both correct -- within the >>>> definitions of eigenvector and eigenvalue they compute, it's OK that >>>> one eigenvector differes between the two by a factor -1. They're >>>> probably calling the same library anyway. >>>> >>>> However, from a physics perspective, this bugs me. In physics, an >>>> important feature of a space is its orientation. For example, >>>> physics and engineering undergrads learn early on about about the >>>> "right-hand rule" for taking cross products of vectors. This type of >>>> product imposes an orientation (or chirality) on the vector space >>>> under consideration. Real mathematicians may be familiar with >>>> the wedge product in exterior algebra, which (I belive) also imposes >>>> an orientation on a vector space. Wikipedia says: >>>> >>>> https://en.wikipedia.org/wiki/Orientation_(vector_space) >>>> >>>> The problem I have with Julia and Matlab returning a set of >>>> eigenvectors where only one eigenvector is the negative of the other >>>> is that the two eigenvector sets span spaces of opposite >>>> orientation, so the returns are -- in some sense -- not the same. >>>> >>>> So my question: Is there anything like a set of "oriented" >>>> eigenvectors? Is this even possible? Or am I totally wrong, and >>>> the two different sets of eigenvectors span the same space independent >>>> of concerns about orientation? >>>> >>>> >>> >>> I don't think you can define that in a continuous way. >>> In general, if your application relies on certain property of the basis, >>> you should just normalize it that way. If you don't have a requirement >>> than >>> you should worry about it. >>> >>> >>> >>>> Stuart >>>> >>>> >>>> >>>> >>>> On Sat, 10 Sep 2016, Tracy Wadleigh wrote: >>>> >>>> Looks good to me. The eigenvalues look the same up to the precision >>>> shown >>>> >>>>> and eigenvectors are only unique up to a scalar multiple. >>>>> >>>>> On Sep 10, 2016 8:11 AM, "Dennis Eckmeier" < >>>>> dennis.eckme...@neuro.fchampalimaud.org> wrote: >>>>> >>>>> Hi! >>>>> >>>>>> >>>>>> Here is a simple example: >>>>>> >>>>>> *Matlab:* >>>>>> (all types are double) >>>>>> covX(1:4,1:4) >>>>>> >>>>>> 1.0e+006 * >>>>>> >>>>>> 3.8626 3.4157 2.4049 1.2403 >>>>>> 3.4157 3.7375 3.3395 2.3899 >>>>>> 2.4049 3.3395 3.7372 3.4033 >>>>>> 1.2403 2.3899 3.4033 3.8548 >>>>>> >>>>>> [V,D] = eig(covX(1:4,1:4)) >>>>>> >>>>>> V = >>>>>> >>>>>> 0.2583 0.5402 -0.6576 0.4571 >>>>>> -0.6622 -0.4518 -0.2557 0.5404 >>>>>> 0.6565 -0.4604 0.2552 0.5403 >>>>>> -0.2525 0.5404 0.6611 0.4551 >>>>>> >>>>>> >>>>>> D = >>>>>> >>>>>> 1.0e+007 * >>>>>> >>>>>> 0.0006 0 0 0 >>>>>> 0 0.0197 0 0 >>>>>> 0 0 0.3010 0 >>>>>> 0 0 0 1.1978 >>>>>> >>>>>> >>>>>> >>>>>> *Julia:* >>>>>> (all types are Float 64) >>>>>> *D = * >>>>>> *5759.7016...* >>>>>> *197430.4962...* >>>>>> *3.0104794 ... e6* >>>>>> *1.1978 ... e7* >>>>>> >>>>>> V = >>>>>> 0.2583 0.5402 -0.6576 *-0.4571* >>>>>> -0.6622 -0.4518 -0.2557 *-0.5404* >>>>>> 0.6565 -0.4604 0.2552 *-0.5403* >>>>>> -0.2525 0.5404 0.6611 *-0.4551* >>>>>> >>>>>> cheers, >>>>>> >>>>>> D >>>>>> >>>>>> >>>>>> On Saturday, September 10, 2016 at 12:48:17 PM UTC+1, Michele Zaffalon >>>>>> wrote: >>>>>> >>>>>> >>>>>>> You mean they are not simply permuted? Can you report the MATLAB and >>>>>>> Julia results in the two cases for a small covX matrix? >>>>>>> >>>>>>> On Sat, Sep 10, 2016 at 1:34 PM, Dennis Eckmeier < >>>>>>> dennis....@neuro.fchampalimaud.org> wrote: >>>>>>> >>>>>>> Hi, >>>>>>> >>>>>>>> >>>>>>>> I am new to Julia and rather lay in math. For practice, I am >>>>>>>> translating >>>>>>>> a Matlab script (written by somebody else) and compare the result of >>>>>>>> each >>>>>>>> step in Matlab and Julia. >>>>>>>> >>>>>>>> The Matlab script uses [V,D] = eig(covX); >>>>>>>> >>>>>>>> which I translated to Julia as: (D,V) = eig(covX) >>>>>>>> >>>>>>>> However, the outcomes don't match, although covX is the same. >>>>>>>> >>>>>>>> D. >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>> >>>>> >>> >
