Talk:Block LU decomposition

A-star over two

I haven't seen the $A^{* \over 2}$ matrix before, is it possible to see more on this matrix than what is on this page. Does it have a name? Pdbailey (talk) 14:18, 2 June 2008 (UTC)[reply]

LDU decomposition clearer?

{\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I&0\\CA^{-1}&I\end{pmatrix}}{\begin{pmatrix}A&0\\0&D-CA^{-1}B\end{pmatrix}}{\begin{pmatrix}I&A^{-1}B\\0&I\end{pmatrix}},

I have always seen it written this way. I'm not sure where the half matrices come in since we need symmetry for that anyway (this article is really showing a block Cholesky factorization, it never has a general LU decomposition). Also decomposing as a sum doesn't help if we are interested in the inverse. I think this LDU form is the most useful, both for analysis and numerically.

JedKBrown (talk) 13:35, 6 September 2008 (UTC)[reply]

Problem

I second what JedKBrown says, the article isn't correct, it only works for positive matrices where it amounts to a block Cholesky. The general case is indeed very similar, just replace cholesky by LU... still that needs to be mentioned.

(edited by Benoit Jacob)

I have tried to add a (non-Cholesky) LU decomposition to the page. Alas my edit was reverted. I don't know why. I also added sample code for how to make use of this. Maybe the bot did not like my code? Thethirdrock (talk) 05:49, 23 December 2014 (UTC)[reply]

Half matrices

I'm not familiar with that term. There isn't a wiki page on it or anything. Can anyone point to anything which explains it? Or just explain it quickly here? --Numsgil (talk) 22:55, 23 July 2009 (UTC)[reply]

It looks like they are defined on the page implicitly. But it would be nice to see the ref. PDBailey (talk) 03:50, 24 July 2009 (UTC)[reply]

Yeah, I saw the definitions in page like that, and I went through the algebra and it all works, I just don't quite understand how you would find the half matrix. Is it basically like the lower triangular matrix you'd get from Cholesky decomposition? —Preceding unsigned comment added by Numsgil (talk • contribs) 16:35, 24 July 2009 (UTC)[reply]

Rewriting the article

So I went through the article and looked for some sources on it. While everything seems to be correct as far as I can tell, the sources that I turned up used different notation that I found more accessible. In particular I ran into: https://arxiv.org/pdf/2201.00145 and https://ntrs.nasa.gov/api/citations/19950017172/downloads/19950017172.pdf I personally would rewrite the article to be more extensive in its explanations, bring notation in line with the sources listed above, and I would add the current notation and proof as a separate section labeled something like "Proof using Half matrices", with a more lengthy explanation of what goes on maybe. Unless anyone objects I would rewrite the article in the manner outlined above. Mangomarkus (talk) 14:42, 24 March 2025 (UTC)[reply]

Also it does not mention that the Cholesky decomposition only holds for PSD matrices (and furthermore specifically only for symmetric matrices.)

The Decomposition that is listed here and labeled "Cholesky" decomposition, seems to just be a specific version of a proof of a Block LU decomposition. Mangomarkus (talk) 15:01, 24 March 2025 (UTC)[reply]