Talk:Definite matrix

message: The statements about the complex/real case were incorrect. Here's a simple counterexample:
\begin{bmatrix} 1 & 2i \\ -2i & 1 \end{bmatrix}
This Hermitian matrix has $x^T A x > 0$ for all real $x$, but not for $x = [1 , i]$.
----
Would it make any difference if x was in Cn?
shd: In that case it should be something like x*Mx where * refers
to complex conjugate transpose
:I added that to the article. AxelBoldt
Does this encompass all positive-definite matrices? The Mathworld page leads me to believe that there are positive-definite matrices that are not Hermitian. From the page: 'A necessary and sufficient condition for a complex matrix to be positive definite is that the Hermitian part ... be positive definite.' inferno
If you think about xtMx as a quadratic form in the vector x, and write the square matric M as a sum of a symmetric part M1 and an antisymmetric part M2, you see that the quadratic form is independent of M2.  So if you look at a symmetric positive definite  M1, you can add any  M2 you want, and still be positive definite. If that's useful ... And the same thing if you split up into a hermitian and anti-hermitian part, for complex x and introducing complex conjugation in the form.
Charles Matthews 20:11, 16 Jun 2004 (UTC)
replies: text: Why does a PD matrix A need to be Hermitian (or symmetric in real case)? This is very confusing and serious. eakbas 04:24, 2 October 2007 (UTC)

text: I'm afraid I can't find the link. The first sentence in Niemeier lattices links to positive definite, which redirect to definite bilinear form. However, the best explanantion of "positive definite" in this context can be found on Lattice (group). -- Jitse Niesen 12:18, 20 Jun 2005 (UTC)
thread_title:  Old remarks 

message: This is all very nice and I'm sure a "rigorous definition" is great for some people.  However, for someone who just got redirected from positive-semidefinite and doesn't have a degree in pure math, this is not very usefull.
Agreed - can someone with a better understanding of the subject please try to give a more intuitive description somewhere?  70.93.249.46
:I may be a mathematician, but I have no idea what could be made clearer. Of course, you'll likely encounter only real matrices for which the definition is simply that B is symmetric and x^T B x >= 0 for all vectors x. Geometrically, the set of n x n positive semidefinite matrices is a cone (you can add two together and multiply them with a nonnegative number), but that's rather trivial, and the definition of the barrier function that makes them useful in optimization is not helpful either. What sort of "useful" facts would you like to learn? Wandrer2 16:35, 2 March 2006 (UTC)
replies: text: I've just expanded property 2 in the definition to give a spectral intuition about how positive definite M are special within the set of Hermitian operators.  For me, this intuition is fundamental and was not quite there before.  Hope it helps.  Eclecticos 08:02, 20 September 2006 (UTC)
thread_title:  English 

message: What is meant by "A positive definite if and only if all eigenvalues are positive"?. Is all eigenvalues >0 or is all eigenvalues \geq0?
I have managed to prove the following:
Let $A$ be a positive definite  n\times n -matrix with eigenvalues \lambda_1,\lambda_2,\ldots,\lambda_n
then
\lambda_i\geq0, \quad i=1,2,3\ldots,n and there exist a k\in\{1,2,3\ldots,n\} such that \lambda_k>0.
But I havn't managed to prove the following:
Let $A$ be a positive definite $n\times n$-matrix with eigenvalues \lambda_1,\lambda_2,\ldots,\lambda_n then
\lambda_i>0,\quad i=1,2,3,\ldots,n.
To be clearer
A=
\begin{pmatrix}
3&0&0\\
0&2&0\\
0&0&0\\
\end{pmatrix}
has the eigenvalues \lambda_1=3,\lambda_2=2,\lambda_3=0.
The matix is positive definite since
x^tAx=\sum_{i=1}^3\lambda_ix_i^2>0
for all vectors x=(x_1,x_2,x_3)\in\mathbf{R^3},x\neq0.
Therefore A symmetric and positive definite doesn't imply that all eigenvalues of $A$ is positive (in the sence >0).
However maybe this might just be the case when the matrix contain a row $j$ and column $j$ that are both zerovectors.
Can anybody help me??? I don't get it.
/Tobias mathstudent
:"A positive definite if and only if all eigenvalues are positive" means that all eigenvalues have to be > 0.
:Your matrix A is not positive definite, because xTAx = 0 for x = (0,0,1), which is not the zero vector. 
:I hope this clarifies the matter. If not, feel free to ask. -- Jitse Niesen (talk) 10:53, 16 August 2006 (UTC)
thread_title: Positive Eigenvalue? 

message: Recall the property mentioned:
9. 	If M > 0 is real, then there is a δ > 0 such that M\geq \delta I where I is the identity matrix.
I guess, we can as well write M > \delta I (because of the eigenvalue characterization), right?
146.186.132.163 19:43, 12 March 2007 (UTC)
replies: text: yes. "is real" presumably means symmetric with real entries. it's true for positive matrices with complex entries also. Mct mht 22:18, 12 March 2007 (UTC)
thread_title:  minor question 

message: This article could do with a diagram showing examples and counter-examples of the range of positive-definite matrices. In particular, I am thinking about 2D transformations that can be represented as ellipses. 155.212.242.34 (talk) 13:00, 13 August 2008 (UTC)
replies: text: 155.212.242.34, sounds like you are farmiliar with this topic. be bold and add the information! Pdbailey (talk) 02:48, 14 August 2008 (UTC)
thread_title:  Diagram 

message: Doesn't the product need to look like B B^T or B B^dagger. (CHF (talk) 04:32, 23 October 2008 (UTC)
replies: text: Well, the property is also true for B B^T, in which case we have the Cholesky decomposition. However, this is already mentioned under number 5 in the Characterization section. I think that Property 4 is supposed to be a different property (for any positive-definite matrix M, there exists a positive-definite B such that M = B2). I can see that it's very easy to misunderstand the text, so I tried to clarify the situation. -- Jitse Niesen (talk) 13:47, 29 November 2008 (UTC)
thread_title:  Property 4 

message: As the article stands it reads, "For positive semidefinite matrices, all principal minors have to be non-negative. The leading principal minors alone do not imply positive semidefiniteness, as can be seen from the example" before semidefiniteness has been defined. I tried a couple of ways to move it down but could not get it to work easily. Can anyone else see a good way to do this? I think this should not be in there anyway because it is pretty long and implies that the other conditions are relatively intact for semidefinite matricies. PDBailey (talk) 22:10, 12 March 2009 (UTC)
thread_title:  lets define positive semidefinite before we use it. 

message: I want to add information about the (differential/Riemannian) geometry of the cone of positive definite (real) matrices. ¿Is this
article the right place, considering that the mathematical level must be higher? ¿If not, were to add it?
--Kjetil Halvorsen 02:32, 9 January 2010 (UTC)
thread_title:  Geometry of cone of positive definite matrices 

message: unless I've missed something, the article uses both the dagger and the star symbol for the conjugate transpose - eek!
--Dmack  —Preceding unsigned comment added by 163.1.167.234 (talk) 23:52, 14 March 2010 (UTC)
thread_title:  inconsistent adjoint notation 

message: The first two examples do not explain what makes them positive-definite or not. In particular it does not state how a vector is chosen for multiplication. ᛭ LokiClock (talk) 17:59, 14 June 2010 (UTC)
replies: text: Yes, they do. The first matrix is positive-definite because the associated quadratic form (which is given) is positive on nonzero vectors (for reasons which are also explained). The second is not positive-definite because the associated quadratic form takes negative values on some vectors (one of which is given explicitly). Algebraist 18:04, 14 June 2010 (UTC)
thread_title:  Examples not explanatory 

message: is there any relationship between these two??? 
I feel that positive definite implies all eigen values are positive, am I right?
Jackzhp (talk) 03:00, 12 February 2011 (UTC)
replies: text: Yes, and the determinant is the product of all eigenvalues and so that makes it positive. CHF (talk) 15:37, 22 December 2011 (UTC)
thread_title: relationship between positive definite and positive determinant

message: (I apologize in advance if this had been hashed out previously--I don't see it covered on this page.)  I noticed today that someone had edited the PSD disambiguation page to change Positive-Semidefiniteness to Positive-Semi-Definiteness.  I thought it looked somewhat odd, so I decided to peruse the linked Positive-definite matrix page.  I see that "semidefinite" generally prevails, except in the Further properties section where both appear.  I also see both in this talk page.  I am comfortable with one being standard or both being acceptable, but I don't see any hints to which it may be in the page or in the talk.  I'm guessing that opinions differ.  
(By the way, feel free to edit this section or paragraph (including heading) mercilessly if it supports the discussion.  You won't hurt my feelings.  My contributions are intended to further the discussion and no attribution is required--if you change it enough, you may want to strike this parenthetical (with my name).  Cheers! rs2 (talk) 03:03, 29 June 2011 (UTC)
replies: text: I strongly agree that consistency would be nice, and "semidefinite" seems to be more popular than "semi-definite".  (FWIW, I also think the unhyphenated version it looks better.)  I'm in favor of changing every instance of the hyphenated version to the unhyphenated version. --Joel B. Lewis (talk) 00:58, 3 July 2011 (UTC)
thread_title: Spelling: Semi-definite vs. Semidefinite?

message: The matrix M3 = {{1 2} {0 1}} in 'Examples' is not positive-definite (e. g. for z = {1 -1}) Sconden (talk) 19:53, 15 May 2012 (UTC)
thread_title:  minor error 

message: The lead paragraph of a Wikipedia article must define the topic, not merely say some vague nice somethings about it. (If there are many competing defintions or subtle details, it should give best or most common definition that can fit in one paragraph, and try to warn the reader about those subtleties.)Moreover, the lead section must define all topics that are redirected to the article -- in this case, "negative definite", "positive semidefinite", and "negative semidefinite".All the best, --Jorge Stolfi (talk) 00:33, 27 July 2012 (UTC)
thread_title:  The lead paragraph must ''define'' the topics 

message: Can somebody tell me where I go wrong here - A real Hermitean matrix is symmetric, so a real positive definite matrix must be symmetric. If so, why is  m={{2,2},{0,1}} not positive definite? I can find no r={x,y} such that r.(m.r)<=0. PAR (talk) 19:51, 20 January 2013 (UTC)
replies: text: Thanks - I have edited the introduction accordingly. PAR (talk) 01:17, 13 August 2013 (UTC)
thread_title:  Positive definite real is symmetric? 

message: The symmetric matrix
M = \begin{bmatrix} 1 & -0.8 & -0.1\\ -0.8 & 1 & -0.8\\-0.1 & -0.8 & 1\end{bmatrix} 
has the required from (with m(0)=1, m(1)=-0.8, m(2)=-0.1) and satisfies
\vert m(1)\vert + \vert m(2)\vert = 0.9 < 1 = m(0)
but is not positive definite. 
141.34.29.108 (talk) 21:34, 6 February 2015 (UTC)
replies: text: For the second row, \vert m(-1)\vert + \vert m(1)\vert = 1.6 > m(0) PAR (talk) 13:34, 26 August 2016 (UTC)
thread_title:  Property #8 incorrect? 

message: The identity matrix is not only positive-semidefinite but also positive definite (all its eigenvalues are >0).
While what is written there is not wrong it would be very confusing for somebody reading this for the first time,
because you might ask why only the weaker statement is given.
Also in this example section a matrix N is mentioned which is never given. Ben300694 (talk) 12:47, 2 March 2017 (UTC)
thread_title:  Improvements needed in the example section 

message: It says: "If M ≥ N > 0 then (...) by the min-max theorem, the kth largest eigenvalue of M is greater than the kth largest eigenvalue of N."
I can't see, how this comes by the min-max theorem and since I can't find this property in any textbook, I am wondering if it's correct. Can anybody provide a proof on that or a reference? BBC89 (talk) 10:44, 4 July 2018 (UTC)
thread_title:  Reference or proof to Further properties #1 

message: In the article I read under Simultaneous diagonalization I read the statement: "X^T M A = \Lambda". I am almost certain that the following is meant: "X^T M X = \Lambda". Can someone well-versed in matrix algebra confirm this?Redav (talk) 18:12, 27 January 2019 (UTC)
replies: text: My fault. Fixed it. Fvultier (talk) 18:40, 27 January 2019 (UTC)
thread_title:  Typo? 

message: A discussion is taking place to address the redirect :Negative semi-definite. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 June 9#Negative semi-definite until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 (talk) 14:36, 9 June 2020 (UTC)
thread_title:  "Negative semi-definite" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: An IP user is edit warring for adding the the lead a confusing comment to the lead of the article. These edits are not aceptable, as consisting essentially of an alleged proof that a matrix that has been supposed to be Hermitian is indeed Hermitian (circular reasoning). If the user continue this way, I'll asking for an edit block per wP:3RR. D.Lazard (talk) 15:04, 2 May 2021 (UTC)
thread_title:  IP-user edits 

message: Hello all,
In the intro there are various conditions for a matrix to be Definite, of the form:
> M is congruent with a diagonal matrix with positive (resp. nonnegative) real entries.
What does *resp.* mean here? Can somebody clarify?
(AFAIK positive and nonnegative are not equivalent, this is confusing)
--kibibu (talk) 02:44, 20 September 2021 (UTC)
replies: text: . "resp." means "respectively". However, as used here with many occurences, this is mathematical jargon that should not be used in the lead. D.Lazard (talk) 08:48, 20 September 2021 (UTC)
thread_title:  Clarification on "resp." in intro 

message: The page is displayed like this on my browser. Is it because of my problem? I don't know how to fix this but it doesn't display like this on others' computers.  Plurm (talk) 01:48, 3 June 2023 (UTC)
replies: text: It is fixed now. Thanks. Plurm (talk) 05:00, 4 June 2023 (UTC)
thread_title:  Displaying problem