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From: Ted Dunning <ted.dunning@gmail.com>
Date: Mon, 6 Jan 2014 22:57:13 -0800
Message-ID: 
 <CAJwFCa3miP_U8E=xBiRYDEBz4-Cg1sNTNEJybGJ4AHsfv9NJzg@mail.gmail.com>
Subject: Re: eigenvectors and eigenvalues of a matrix
To: "user@mahout.apache.org" <user@mahout.apache.org>
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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
>

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