Talk:Cholesky decomposition

message: In the Monte Carlo example application of Cholesky decomposition, \rho is never explained  — Preceding unsigned comment added by Matrixalgebrauser (talk • contribs) 04:45, 28 February 2014 (UTC)
thread_title:  Lack of definition of in Monte Carlo example 

message: These two terms are not defined anywhere in Wikipedia, and searching on the web turns up few references. This terminology is not common, and should be explained.Eigenguy (talk) 01:04, 25 July 2012 (UTC)
thread_title:  Rank one Update and Downdate 

message: Thanks for the simple explanation Jitse, 
I wasn't sure whether all matrices, A, where v*A.v > 0 were Hermitian or not. 
Isn't that the general definition of PD though? 
Is it specified that A must also be Hermitian to qualify as PD? 
Based on my hazy education, and what I see on Wolfram Research, and Computation Methods in Linear Algebra I am forced to dispute this definition. 
Everything I've seen specifies simply the first condition (the multiplication property) and not the Hermitian property (Planet Math fudges the issue by saying that "If A is a square Hermitian then ... is it a positive definite matrix", without saying that this is the non-Hermitians with the same property are not. I have seen no definitions explicitly specifying this condition.) For instance, the definition Google uses, from University of Waterloo simply says (): 
: Positive definite matrix (A). x'Ax > 0 for all nonzero x. 
Importantly, Computation Methods in Linear Algebra Proposition 6.2.2 clearly implies:
# Being Hermitian is not a condition for a matrix to be PD (because it states that both conditions must be fulfilled for Cholesky factorization to succeed they must be logically independent).
# Therefore, the Cholesky decomposition page must make this condition (symmetry/Hermitian) explicit, as the original questioner requested
I conclude that the original questioner raised a useful & correct gap in the definition, but if I have made a mistake, It would be interesting to see where. Can you tell me why you (Jitse Niesen) think that Matrices, A, where all non-zero v vectors give:
v*A.v > 0 
must also be Hermitian to qualify as a positive-definite matrix?
I only work with symmetric matrices, and most programs and references refer to "Positive Definite Symmetric" as a subset of "Positive Definite", implying that there is no such thing as "Asymmetric Positive Definite Matrix". If such a thing does exist, then some people (me anyway) would be interested in a topic or heading with that name, and an example. It would be one of the few on the Internet. 
Also, I am not a pure mathematician, so please forgive any basic errors in understanding that I may have made. I don't know if this is relevant, but the structure of "Definite bilinear form" has the same "if Hermitian then PD" form, without saying "iff Hermitian then PD". Even if this is right, I think it should be cleared up.
- Wragge 22:19, 2005 Apr 28 (UTC)
:I pulled some books from the shelf and they disagree on this issue. Most use a formulation like "A Hermitian matrix A is positive definite if v*Av > 0," which is ambiguous. Some imply further on that they consider being Hermitian to be a condition for a matrix to be positive definite and some imply that Hermitian and positive definite are independent properties. So, I think it is safest to stress in this article that we need A to be both Hermitian and positive definite. I wrote a small discussion over the relation between positive definite and Hermitian at Positive-definite matrix#Non-Hermitian matrices. -- Jitse Niesen 21:33, 29 Apr 2005 (UTC)
thread_title:  PD MUST BE HERMITIAN IN DISPUTE 

message: Question:
The page says that only a positive-definite matrix may be factorized as seen. Are positive-SEMIdefinite matrices excluded from this factorization?
::I added a paragraph to address this question. -- Jitse Niesen (talk) 04:37, 10 May 2006 (UTC)
replies: text: In that case, the Cholesky–Banachiewicz algorithm has a flaw for semidefinite matrices. To compute L_{i,j} below the diagonal, it divides by L_{j,j}, which could be zero in the semidefinite case. I have seen code that checks for this and just sets the cell to zero, but I am not terribly confident that this is the right way to handle it. Anyone know? --Dave.rudolf (talk) 18:32, 24 January 2011 (UTC)
replies: text: Quick search found the article "Analysis of the Cholesky Decomposition of a Semi-deﬁnite Matrix" analyzing the numerical stability of Cholesky decomposition for symmetric positive-definite matrices with pivoting. I only read the beginning, but it refers to LINPACK routine SCHDC from 1979 which does it, so algorithms for the semidefinite case are not new (the numerical stability analysis article is from 2008, though!). It would be nice if the Wikipedia article treated the semidefinite case as well. -- Coffee2theorems (talk) 10:29, 28 July 2012 (UTC)
replies: text: I believe that the proof given for the positive semi-definite case has a flaw, and I think several simplifications can be made in other areas. First, regarding the part which says "by property of the operator norm", I do not see which property is being used, and in fact I believe that the inequality is the opposite way. Second, to show A = LL^*, we do not need to go through inner products.Suncup224 (talk) 10:36, 30 December 2013 (UTC)

text: The introduction of the article says positive-definite, the statement paragraph state positive-semidefinite. We should be consistent with the definition Raffamaiden (talk) 16:02, 1 May 2013 (UTC)
replies: text: I checked three references and all of them define initially the Cholesky decomposition only for positive-definite matrices and they're not clear on whether the name also applied to positive-semidefinite. I changed the statement paragraph to be closer to the references. -- Jitse Niesen (talk) 15:20, 2 May 2013 (UTC)
thread_title:  The semidefinite case 

message: I belive the formulas given for the decomposition relate only to real (real SPD) case. I think they should be generalised to the complex Hermitian positive-definite case. Than they will look like that:
l_{ii} = \sqrt{a_{ii} - \sum_{k = 1}^{i - 1}{|l_{ik}|^2}}
l_{ij} = \frac{1}{l_{jj}} \left( a_{ij} - \sum_{k = 1}^{i - 1}{l_{ik}\bar{l}_{jk}} \right)
One more question came to my mind...Why do we consider only positive-definite matrices in complex case??? There exists square root for any complex number (not unique however). So we should be able to use above formulas for any Hermitian matrix (not necessarily PD). Of course the diagonal elements of L would not be positive reals anymore. 
What is more, we should be able to use the original formulas for symmetric real case in symmetric complex case. So why do we care about positive-definitness???
--A-lbi 09:49, 22 August 2007 (UTC)
thread_title:  The real matrix vs. complex matrix decomposition 

message: As someone who doesn't know much about mathematics, could there be a definition (or a link to a definition) on what is meant by pivot in the sentence "Any square matrix A with non-zero pivots can be written as the product of..."? Please forgive my ignorance but hey - that's why I'm reading the article in the first place ;-) —Preceding unsigned comment added by 158.234.250.71 (talk • contribs) 16:30, 1 July 2008 
Bump for this one: it would be useful to have an description of the pivoting procedure.  — Preceding unsigned comment added by 128.125.86.205 (talk) 20:35, 5 August 2013 (UTC)
thread_title:  Pivot? 

message: I added a numerical recipes cite in the first paragraph and changed the sentence to read "roughly half" instead of "half [as many operations]". It is still possible that Cholesky decomposition requires literally half as many operations but we should cite a source before saying so. I found several other sources that imply it is but that never say specifically. Jason Quinn (talk) 13:07, 29 January 2009 (UTC)
replies: text: Trefethen and Bau (Numeical Linear Algebra, 1997) say it is 2/3 m3 flops for LU (pp 152) and 1/3 m3 flops for Cholelsy. PDBailey (talk) 00:24, 30 January 2009 (UTC)
replies: text: They ignore lower order terms, though. I think it would be quite a coincidence if the number of operations were exactly halved. -- Jitse Niesen (talk) 13:36, 30 January 2009 (UTC)

text: You should add the Trefeten reference to the article. Probably a large segment the of the readership of this article will be interested in the big-O order. Jason Quinn (talk) 17:20, 30 January 2009 (UTC)

text: How does it make any sense to talk about FLOPS? How many steps an algorithm takes has nothing to do with either floating point operations nor how many are performed per second. Giving the order of the number of steps in Landau notation doesn't really help as a comparison to LU decomposition either.Steve Checkoway (talk) 23:11, 24 January 2011 (UTC)
thread_title:  BigO of Cholesky vs LU? 

message: The hyphenated terminology -Banachiewicz and -Crout does not exist in numerical analysis.  The use of these phrases is an attempt by someone to create a historical reference that is not appropriate.  I sympathize with whoever was the author, because Banachiewicz and Crout did have something to do with solving linear systems and deserve more credit.  However. if that history is wanted, a proper history section should be added rather than the made-up names. :-)  Jfgrcar (talk) 02:21, 26 September 2009 (UTC)
thread_title:  The Cholesky-Banachiewicz and Cholesky-Crout algorithms ?? 

message: With some web searching, I found many people asking why we need the positive definite restriction in the complex (Hermitian) case, but didn't find anyone attempting an answer.  In the real case, the algorithm gives us a way to detecting non-positive-definiteness, but how do you do that in the complex case?
I notice that in the complex case, you end up with:
l_{ii} l_{ii}^* = a_{ii} - \sum_{k = 1}^{i - 1}{|l_{ik}|^2}
If the right-hand side is not positive real, I don't think you can find l_{ii}.  Notice that it isn't a straight square root, because the diagonal term times its conjugate must be equal to the right-hand side, as opposed to the term times itself.
Do either or both of these conjectures hold (when a is Hermitian)?:
* if a is not positive definite, then a_{ii} - \sum_{k = 1}^{i - 1}{|l_{ik}|^2} \le 0 for some i.
* If a_{ii} - \sum_{k = 1}^{i - 1}{|l_{ik}|^2} \le 0 for some i, then a is not positive definite.  —Preceding unsigned comment added by 67.160.223.41 (talk) 23:02, 13 December 2009 (UTC)
thread_title:  Hermitian case when not pos-definite 

message: I know that many built in functions into programs to compute this decomposition use an upper triangular formulation. Although lower triangular is definitely more classic, isn't the upper triangular version ie. XtX = UUt also a cholesky decomposition MATThematical (talk) 23:18, 14 March 2010 (UTC)
thread_title:  Upper triangular 

message: I understand intuition of eigendecomposition of a Hermetian matrix (for a covariance, it's the variances in principal directions; for a coordinate transformation it is similar). What is the intuitive, physical meaning of Cholesky decomposition? There must be some interpretation...? —Ben FrantzDale (talk) 05:05, 19 April 2010 (UTC)
thread_title:  Intuition? 

message: wouldn't you need to the square root matrix of the inverse of the Hessian, not the square root of the Hessian? 151.200.188.223 (talk) 06:42, 26 February 2011 (UTC)
thread_title:  for MC simulation 

message: In the explanation of the cholesky algorithm, I think the part that relates A(i) to A(i+1) is impossible to understand without knowing how B(i+1) is related to B(i). It feels like something is missing  — Preceding unsigned comment added by 83.251.122.201 (talk) 21:11, 15 August 2012 (UTC)
thread_title:  The cholesky algorithm 

message: it is said that "L is a lower triangular matrix with nonnegative diagonal entries". L in general is a mtrix with complex numbers. What does it mean that the ocmplex numbers that are on the diagonal are non-negative?  — Preceding unsigned comment added by Raffamaiden (talk • contribs) 16:07, 1 May 2013 (UTC)
replies: text:  The diagonal entries of L are real. Here is an argument. A is hermitian positive definite, so it has real eigenvalues and is unitarily diagonalizable, and thus has a Hermitian positive definite matrix square root; if A = U D U^H, then S = A^{1/2} = U D^{1/2} U^H, where D^{1/2} has (real) positive entries. S in turn has a QR decomposition S = QR. R is upper triangular with real diagonal entries (the real eigenvalues of S). We have A = S^2 = S^TS = R^H Q^H Q R = LL^H, where L = R^H and thus has real diagonal entries. The cholesky decomposition is unique, so this construction serves to show that the diagonal entries of L are real. Khromegnome (talk) 21:07, 1 May 2013 (UTC)
thread_title:  negative complex number 

message: It states
:\| \mathbf{L}_k \|^2 \le \|  \mathbf{L}_k \mathbf{L}_k ^* \|.
The operator norm is submultiplicative, why should this be the case?
--Erzbischof (talk) 11:14, 9 July 2013 (UTC)
replies: text: The case of \mathbf{L}_k \mathbf{L}_k ^* is special.  For example, in the induced L2 norm (i.e. the operator norm induced by the L2 norm on the vector space), then | A |^2 = |  AA^* |: | A |^2 is the square of the maximum singular value of A and this is the same as the maximum singular value of AA^*.   I'm not sure off the top of my head about other induced norms, but it seems plausible (which doesn't obviate the need for a reference).  Indeed, the whole section lacks references. — Steven G. Johnson (talk) 14:23, 9 July 2013 (UTC)
replies: text:  Ah, thank you, so for the spectral norm it holds and then norm equivalence gives a bound anyway. On the other hand, | A |^2 = |  AA^* | look like a Hilbert space property. --Erzbischof (talk) 21:37, 9 July 2013 (UTC)
thread_title:  [[Cholesky_decomposition#Proof_for_positive_semi-definite_matrices]] 

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