| Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices |
[ Thread Index |
Date Index
| More lists.tuxfamily.org/eigen Archives
]
- To: eigen@xxxxxxxxxxxxxxxxxxx
- Subject: Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices
- From: Ben Goodrich <bgokgm@xxxxxxxxxxxxxx>
- Date: Wed, 24 Feb 2010 09:45:20 -0500
- Dkim-signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlemail.com; s=gamma; h=domainkey-signature:mime-version:received:in-reply-to:references :date:message-id:subject:from:to:content-type :content-transfer-encoding; bh=JgQZ6xV9sVtOHTp70kn11E/QNiHmfTfM7hIOwCXNY1U=; b=DUN2ldMkeqFDVu+TT0haq6v4wW5zKFwzUV885l3oHYZUHVugyt64FHXETXLU6PF1Gy 8m5itinxfulaplqhpR4FbpMY3u3UjNzimRljCEqDf9QDJH/bmJpz+Di6tvkbm1PC+ghN e3+7JBfwITWkO8nu8pEpEZZLYBuSkxDXfwlyk=
- Domainkey-signature: a=rsa-sha1; c=nofws; d=googlemail.com; s=gamma; h=mime-version:in-reply-to:references:date:message-id:subject:from:to :content-type:content-transfer-encoding; b=OjDmEZDnTkJIIz4QPRF4WtMkzIOqUy0f/3LRzYts6usaSVns9Ataat1qx6SrYvrbJO PmXXZ9TaM78ILzMJ6nbYSHz+wx46CcGlxs/uNke5AQkRCpO2FHm0hA+ynnil0k2Dl4u9 s0dtEQ0Gr2graWi0+6O7dpKoBLnM5pEQhL6go=
On Wed, Feb 24, 2010 at 5:53 AM, Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx> wrote:
> On Wed, 24 Feb 2010, Gael Guennebaud wrote:
>
>> note that the LDLT decomp does pivoting, so it is normal that you don't
>> get the same diagonal matrix. If you reconstruct the matrix from the decomp
>> you will see that in this case the decomp is correct.
>
> To further clarify, because this also threw me off for a while: I think the
> important thing here is that the matrix A is singular. The diagonal D is
> unique (up to reordering) for positive-definite matrices. However, there is
> no such uniqueness for singular matrices. Simple example
>
> A = [ 0 0; 0 1 ] (these are 2-by-2 matrices in Matlab notation)
> L = [ 0 0; sqrt(3) 1 ]
> D = [ 1/4 0; 0 1/4 ]
>
> The matrix A has two LDL^T decompositions, A = A * I * A^T = L * D * L^T,
> but the diagonal parts are not the same.
>
> Cheers,
> Jitse
On pages 12 -- 13 (section 5.3) of this paper by Benson and Vanderbei
http://www.pages.drexel.edu/~hvb22/dimacsppr3.pdf
they claim (without proof) that D is unique in the positive
semi-definite case (although part of L is not when A is singular).
Maybe they are incorrect, but I don't think your example shows that
because
L = [ 0 0; sqrt(3) 1 ]
is not *unit* lower triangular. If we restrict L to be unit lower
triangular, then in your example the (a?) LDL' decomposition of A is
L = I and A = D.
So, I am still confused :) but thanks to everyone for their comments
and to Gael for implementing reconstructedMatrix() .
Ben