Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices |

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*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

**Follow-Ups**:**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Jitse Niesen

**References**:**[eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Benoit Jacob

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Benoit Jacob

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Gael Guennebaud

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Jitse Niesen

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