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From Tharindu Rusira <tharindurus...@gmail.com>
Subject Re: eigenvectors and eigenvalues of a matrix
Date Wed, 08 Jan 2014 12:05:26 GMT
On Wed, Jan 8, 2014 at 4:58 PM, Sebastian Schelter <ssc@apache.org> wrote:

> I would rather use a List of Pair<Double,Vector> entries then a HashMap
> with the eigenvalue as key.

Sure Sebastian, it's a better design. I wanted to verify that the line
eigenMap.put(singularValues[i] * singularValues[i], (DenseVector)
eigenVectors.viewRow(i)); is correct.

Thanks,

HashMap don't have an ordering and you will
> get problems if you have to eigenvalues of the same magnitude.
>
>
> On 08.01.2014 12:13, Tharindu Rusira wrote:
> > Hi Ted, Thank a lot for your comprehensive reply.
> >
> > The following is a simple piece of code I wrote. According to what you
> have
> > said, this code would make sense, am I right? Please correct me if I'm
> > wrong.
> >
> > /*respective parameters of the following method are taken from
> > getSingularvalues() and getV() methods in SingularValueDecomposition
> class.
> > */
> >
> > public HashMap<Double,DenseVector> mapEigenvaluesToEigenVectors(double[]
> > singularValues, Matrix eigenVectors) {
> >
> >     HashMap<Double,DenseVector> eigenMap = new
> > HashMap<>(singularValues.length);
> >
> >     for (int i = 0; i < singularValues.length; i++) {
> >
> >       // eigenvalue = singular value ^2
> >
> >       // eigenvectors = right singular vectors
> >
> >       eigenMap.put(singularValues[i] * singularValues[i], (DenseVector)
> > eigenVectors.viewRow(i));
> >
> >     }
> >
> >     return eigenMap;
> >
> >   }
> >
> >
> > In case my problem is unclear, here's some context,
> >
> > I have an input matrix X and I want to find eigenvalues/eigenvectors of
> the
> > covariance matrix XTX. So my workaround is to find singular values and
> > right singular vector of X in order to use the following equivalency.
> > " Comparison with the eigenvector factorisation of *X*T*X* establishes
> that
> > the right singular vectors *W* of *X* are equivalent to the eigenvectors
> of
> > *X*T*X*, while the singular values *σ*(*k*) of *X* are equal to the
> square
> > roots of the eigenvalues *λ*(*k*) of *X*T*X*. " ~ Wikipedia [1]
> >
> > [1]
> >
> http://en.wikipedia.org/wiki/Principal_component_analysis#Singular_value_decomposition
> >
> >
> > Regards,
> >
> >
> >
> > On Tue, Jan 7, 2014 at 12:27 PM, Ted Dunning <ted.dunning@gmail.com>
> wrote:
> >
> >> The order of the singular values and vectors should tell you.
> >>
> >> For others who might be curious, the singular value decomposition
> breaks a
> >> matrix A into three factors
> >>
> >>     A = U S V'
> >>
> >> Both U and V are orthonormal so that U' U = I and V' V = I.  S is
> diagonal.
> >>
> >> An eigenvalue decomposition decomposes a square matrix into two parts,
> one
> >> repeated
> >>
> >>     A = U S U*
> >>
> >> If A is symmetric, then these are real matrices and U* = U'.
> >>
> >> If A is symmetric, we can also decompose it using a Cholesky
> transformation
> >>
> >>     A = R' R
> >>
> >> Where R is upper (right) triangular.  We can then decompose R with SVD.
> >>  This gives us:
> >>
> >>     A = R' R
> >>     R = U S V'
> >>     R' R = (U S V')' (U S V') = V S U' U S V' = V S^2 V'
> >>
> >> a nice convenience is that S^2 is also diagonal and contains the
> elements
> >> of S, just squared.
> >>
> >> So the answer for Tharindu is that the elements of V are not changed or
> >> re-ordered and neither are the elements of S.
> >>
> >>
> >> On Mon, Jan 6, 2014 at 10:22 PM, Tharindu Rusira
> >> <tharindurusira@gmail.com>wrote:
> >>
> >>> Hi,
> >>> I am currently working with SingularValueDecomposition class and I like
> >> to
> >>> clarify the following.
> >>>
> >>> My goal is to find eigenvalues and corresponding eigenvectors of a
> >> matrix.
> >>> I know how to calculate eigenvalues and eigenvectors using svd but is
> >> there
> >>> a way to keep track of which eigenvector corresponds to which
> eigenvalue?
> >>>
> >>> Thanks,
> >>> --
> >>> M.P. Tharindu Rusira Kumara
> >>>
> >>> Department of Computer Science and Engineering,
> >>> University of Moratuwa,
> >>> Sri Lanka.
> >>> +94757033733
> >>> www.tharindu-rusira.blogspot.com
> >>>
> >>
> >
> >
> >
>
>


-- 
M.P. Tharindu Rusira Kumara

Department of Computer Science and Engineering,
University of Moratuwa,
Sri Lanka.
+94757033733
www.tharindu-rusira.blogspot.com

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