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The statements about the complex/real case were incorrect. Here's a simple counterexample:
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
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)
Actually, from the point of view of applications, the restriction of the definition to symmetric matrices (in the real case) is unfortunate, since for instance discretizations of advection problems yield non-symmetric positive definite matrices. I'll change this unless I receive serious complaints. Unfortunately, the article
Normal matrix becomes wrong then. --
Guido Kanschat
20:15, 3 November 2005 (UTC)
This page is linked to from Niemeier lattices. The usage there is a "positive definite unimodular lattice". So we should probably explain here what "positive definite" means in the context of a lattice. I don't know the answer. A5 06:02, 20 Jun 2005 (UTC)
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 completely agree with 70.93.249.46. I have a good background in Maths-for-Natural-Sciences and I currently want to explain what an invertible matrix is to a colleague who also has good MfNS but was never taught about matrices. This page doesn't help either of us very much. Could a link be added to a page about matrix operations at A-level/ US High School/ Baccalaureat level? (Is there one...?) OldSpot61 ( talk) 13:41, 9 April 2008 (UTC)
I would like to prove that the difference between two general matrices (each of a certain class) is a positive semidefinite matrix. I am not up to the task without some examples; would anybody mind posting examples of positive semidefinite (or definite) proofs?
What is meant by "A positive definite if and only if all eigenvalues are positive"?. Is all eigenvalues >0 or is all eigenvalues ?
I have managed to prove the following: Let $A$ be a positive definite -matrix with eigenvalues
then and there exist a such that .
But I havn't managed to prove the following:
Let $A$ be a positive definite $n\times n$-matrix with eigenvalues then
To be clearer has the eigenvalues .
The matix is positive definite since for all vectors .
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
If and are positive definite, then the sum and the products and are also positive definite; and if , then is also positive definite.
Can anyone provide a reference or a sketch of a proof to the part? I have not found it in any googleable literature and I cannot prove it either. Thanks.
Regarding the recent addition by Kkliger, could you provide a reference for this usage? That is, I'm much more familiar with indicating a nonnegative matrix rather than a positive semi-definite one. Feel free to add it back, but it'd be nice if you also note the potential for confusion (and add a reference to support the usage). Cheers, Lunch 19:58, 14 February 2007 (UTC)
kkilger 21:46, 14 February 2007 (UTC)
I've also seen the notation (etc.) used. This is used by Boyd in Convex Optimization, and I think I've seen it elsewhere (systems/control literature), but I can't recall exactly off the top of my head. Whether it's more or less confusing is debatable, though, since for vectors, usually means componentwise greater-than. Overall, it's probably a win, assuming you know what's a matrix and what's a vector. -- Paul Vernaza ( talk) 04:48, 10 February 2008 (UTC)
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)
The leading paragraph states that a positive definite matrix is Hermitian. Isn't that simply wrong? (The same article considers non-Hermitian positive definite matrices, anyway). The last sentence speaks of some "disagreement" about the definition for complex matrices; the article at Mathworld, however, explains it is only that some authors merely restricted the discussion on Hermitian matrices and that the definition have instances in matrices real and complex, Hermitian and non-Hermitian. -- 213.6.23.13 ( talk) 16:07, 24 February 2008 (UTC)
I recently edited the page to remove the mention of requirement that a positive definite (PD) matrix is Hermetian because it is clearly not a requirement (see discussion above). I also put a clear, traditional, definition in the leed so that people would know what we were talking about on this page. I also changed reference of complex numbers with a trailing note about real number to a article that featured reals and had complex numbers mentioned afterwords. the basic idea here being to improve the readability for the audience who knows the least about the subject. Reals are easier to understand, and more people know them. It is my believe that those who understand complex numbers and the concept of a conjugate can easily see how the theorems I changed would work with the complex numbers. Pdbailey ( talk) 04:22, 4 July 2008 (UTC)
(de-indent) As I said, I don't care on whether real or complex is put first. However, I reverted the rest of your edit, because it was quite frankly a mess. Please put more efforts in polishing your edits; this seems to be part of what angered Mct mht and I'm not too happy about it either. -- Jitse Niesen ( talk) 20:01, 16 August 2008 (UTC)
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)
Doesn't the product need to look like B B^T or B B^dagger. ( CHF ( talk) 04:32, 23 October 2008 (UTC))
On the Cholesky decomposition page, it says that the decomposition exists for any positive DEFINITE matrix. On THIS page, it says that the Cholesky decomposition exists for any positive SEMI-definite matrix. Which is it?
Also, I would like to see a section that explains the physical significance of positive definite matrices. It is explained in some haphazard locations that covariance matrices are positive definite, as are the "normal equations" in least squares problems. Why is that? What would it mean if one of those matrices were not positive definite?
In the linear least squares page, it says that "positive definite == full rank", and rank is the number of linearly independent rows. This description is MUCH easier to understand than the stuff on this page about eigenvalues. Is the rank definition correct? —Preceding unsigned comment added by Yahastu ( talk • contribs) 15:08, 20 January 2009 (UTC)
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)
I think the article would be greatly improved if there was a small section after the Definition called Examples showing a simple example of a 2x2 matrix which is positive-definate and an example of a 2x2 matrix which is not. For the latter it would be great to show a single vector which proves this fact. —Preceding unsigned comment added by Pupdike ( talk • contribs) 17:43, 21 March 2009 (UTC)
I went ahead and added the example section. I think it serves to improve the value of the page, but please let me know if you feel otherwise. Pupdike ( talk) 22:08, 23 March 2009 (UTC)
An example of positive definite matrix
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) —Preceding unsigned comment added by Kjetil1001 ( talk • contribs)
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)
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)
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)
(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))
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)
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)
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)
That'd be my question too, more or less. In the section "Quadratic forms" it is said: "It turns out that the matrix M is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function." I would be really surprised that symmetric is a necessary condition for positive definite. In fact, PAR gave the counter example, the symmetric part of [2 2; 0 1] is positive definite as can be seen from its eigenvalues, and so is the matrix itself. But I don't know what the author of that section had in mind... — Preceding
unsigned comment added by
137.226.57.179 (
talk)
09:13, 14 June 2013 (UTC)
This is just a question of convention. Any real square matrix can be written as a sum of a symmetric matrix, , and an antisymmetric matrix, . Then we have the dot product . Since dot products are symmetric, the term involving the antisymmetric matrix always vanishes: , so , which means it must be zero. Because of this, many people do not bother to define positive-definiteness for non-symmetric matrices. In fact, restricting positive-definite to apply only to symmetric matrices means that we can say that a matrix is positive-definite if and only if all its eigenvalues are positive. This statement would not be true if positive-definite matrices were allowed to be non-symmetric. But again, in the end, this is just a question of definition, and the definition in which positive-definiteness implies symmetry seems to be the more common one.
Legendre17 (
talk)
19:25, 12 August 2013 (UTC)
Under characterizations, the eigendecomposition is wrong. I think its supposed to be PDP^-1. If you follow the link "unitary matrix" it says the same (U = PDP* with P* = P^-1). Thus the proof is also wrong (maybe) — Preceding unsigned comment added by 188.155.117.79 ( talk) 21:24, 25 April 2014 (UTC)
I want to see a proof for that property, especially the part that is symmetric.
The symmetric matrix
has the required from (with m(0)=1, m(1)=-0.8, m(2)=-0.1) and satisfies
but is not positive definite. 141.34.29.108 ( talk) 21:34, 6 February 2015 (UTC)
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)
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)
In the article I read under Simultaneous diagonalization I read the statement: "". I am almost certain that the following is meant: "". Can someone well-versed in matrix algebra confirm this? Redav ( talk) 18:12, 27 January 2019 (UTC)
The page contains the following statement:
"More generally, a twice-differentiable real function on real variables has local minimum at arguments if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices."
However, positive semi-definite Hessian doesn't guarantee local minimum, it may also be a saddle point. So either it must be changed to "positive definite", or it may be stated that if a point is local minimum then Hessian must be positive semi-definite.
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)
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)
D.Lazard recently reverted my edits on the intro section which, among other things, defined a positive definite matrix in terms of the angle between and . D.Lazard pointed out that this only applies in Euclidean spaces, so strictly speaking the definition should only be made in terms of inner products. I recognize this point and thank D.Lazard for the correction.
However, I still think that the intro section could use some clean up and clarification and that the rest of my edits were essentially good. I'd also like to point out that the current intro section still has a sentence which asserts that the angle between and is "within an angle of of ," without the clarification that this only applies in Euclidean space.
There are a couple of issues here. First, if we're going to stick with the general outline of the current intro, where positive definite, positive semi-definite, negative definite, and negative semi-definite are all defined separately, then I think it would make sense for the very first sentence to go something like this:
I think it's important to include the connection to right angles in Euclidean space because, at least going from my own experience, the whole concept of (positive) definite matrices only started to make sense to me when I read an explanation that made that connection. At bare minimum, we should point out that is an inner product between and — while that may be kind of obvious to anyone who already understands these concepts, to someone who is new to linear algebra it just looks like a rather arbitrary sequence of matrix-vector multiplications. Since the intro already has a sentence explaining that means the transpose of - a sentence which honestly seems a little unnecessary for me- it seems like we are already on some level committed to making this article accessible for beginners in linear algebra.
Secondly, I think there's an argument to be made that since the title of the article is Definite matrix and not Positive definite matrix, the first sentence should actually be some kind of more general definition of definite matrices, and not a definition of positive definite ones. It might be tricky to come up with an easy-to-understand definition of the general concept before explaining the specific categories, though, so I'd only support this change if we can come up with a really clear definition that flows well into the definitions of positive definite, positive semi-definite, etc. Montgolfière ( talk) 19:45, 4 May 2021 (UTC)
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)
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)
Some of the equivalent conditions in the first section are incorrect. The matrix does not have to be symmetric. See for example https://math.stackexchange.com/questions/1954167/do-positive-semidefinite-matrices-have-to-be-symmetric 158.38.1.98 ( talk) 10:59, 23 May 2024 (UTC)
Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.It seems that you consider these generalizations as a norm and that this is incorrect to not emphasizing them. This is a legitimate opinion, but other editors, including myself, have an opposite opinion. In any case, there is nothing incorrect in the present state of the lead. D.Lazard ( talk) 12:54, 23 May 2024 (UTC)
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The statements about the complex/real case were incorrect. Here's a simple counterexample:
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
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)
Actually, from the point of view of applications, the restriction of the definition to symmetric matrices (in the real case) is unfortunate, since for instance discretizations of advection problems yield non-symmetric positive definite matrices. I'll change this unless I receive serious complaints. Unfortunately, the article
Normal matrix becomes wrong then. --
Guido Kanschat
20:15, 3 November 2005 (UTC)
This page is linked to from Niemeier lattices. The usage there is a "positive definite unimodular lattice". So we should probably explain here what "positive definite" means in the context of a lattice. I don't know the answer. A5 06:02, 20 Jun 2005 (UTC)
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 completely agree with 70.93.249.46. I have a good background in Maths-for-Natural-Sciences and I currently want to explain what an invertible matrix is to a colleague who also has good MfNS but was never taught about matrices. This page doesn't help either of us very much. Could a link be added to a page about matrix operations at A-level/ US High School/ Baccalaureat level? (Is there one...?) OldSpot61 ( talk) 13:41, 9 April 2008 (UTC)
I would like to prove that the difference between two general matrices (each of a certain class) is a positive semidefinite matrix. I am not up to the task without some examples; would anybody mind posting examples of positive semidefinite (or definite) proofs?
What is meant by "A positive definite if and only if all eigenvalues are positive"?. Is all eigenvalues >0 or is all eigenvalues ?
I have managed to prove the following: Let $A$ be a positive definite -matrix with eigenvalues
then and there exist a such that .
But I havn't managed to prove the following:
Let $A$ be a positive definite $n\times n$-matrix with eigenvalues then
To be clearer has the eigenvalues .
The matix is positive definite since for all vectors .
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
If and are positive definite, then the sum and the products and are also positive definite; and if , then is also positive definite.
Can anyone provide a reference or a sketch of a proof to the part? I have not found it in any googleable literature and I cannot prove it either. Thanks.
Regarding the recent addition by Kkliger, could you provide a reference for this usage? That is, I'm much more familiar with indicating a nonnegative matrix rather than a positive semi-definite one. Feel free to add it back, but it'd be nice if you also note the potential for confusion (and add a reference to support the usage). Cheers, Lunch 19:58, 14 February 2007 (UTC)
kkilger 21:46, 14 February 2007 (UTC)
I've also seen the notation (etc.) used. This is used by Boyd in Convex Optimization, and I think I've seen it elsewhere (systems/control literature), but I can't recall exactly off the top of my head. Whether it's more or less confusing is debatable, though, since for vectors, usually means componentwise greater-than. Overall, it's probably a win, assuming you know what's a matrix and what's a vector. -- Paul Vernaza ( talk) 04:48, 10 February 2008 (UTC)
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)
The leading paragraph states that a positive definite matrix is Hermitian. Isn't that simply wrong? (The same article considers non-Hermitian positive definite matrices, anyway). The last sentence speaks of some "disagreement" about the definition for complex matrices; the article at Mathworld, however, explains it is only that some authors merely restricted the discussion on Hermitian matrices and that the definition have instances in matrices real and complex, Hermitian and non-Hermitian. -- 213.6.23.13 ( talk) 16:07, 24 February 2008 (UTC)
I recently edited the page to remove the mention of requirement that a positive definite (PD) matrix is Hermetian because it is clearly not a requirement (see discussion above). I also put a clear, traditional, definition in the leed so that people would know what we were talking about on this page. I also changed reference of complex numbers with a trailing note about real number to a article that featured reals and had complex numbers mentioned afterwords. the basic idea here being to improve the readability for the audience who knows the least about the subject. Reals are easier to understand, and more people know them. It is my believe that those who understand complex numbers and the concept of a conjugate can easily see how the theorems I changed would work with the complex numbers. Pdbailey ( talk) 04:22, 4 July 2008 (UTC)
(de-indent) As I said, I don't care on whether real or complex is put first. However, I reverted the rest of your edit, because it was quite frankly a mess. Please put more efforts in polishing your edits; this seems to be part of what angered Mct mht and I'm not too happy about it either. -- Jitse Niesen ( talk) 20:01, 16 August 2008 (UTC)
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)
Doesn't the product need to look like B B^T or B B^dagger. ( CHF ( talk) 04:32, 23 October 2008 (UTC))
On the Cholesky decomposition page, it says that the decomposition exists for any positive DEFINITE matrix. On THIS page, it says that the Cholesky decomposition exists for any positive SEMI-definite matrix. Which is it?
Also, I would like to see a section that explains the physical significance of positive definite matrices. It is explained in some haphazard locations that covariance matrices are positive definite, as are the "normal equations" in least squares problems. Why is that? What would it mean if one of those matrices were not positive definite?
In the linear least squares page, it says that "positive definite == full rank", and rank is the number of linearly independent rows. This description is MUCH easier to understand than the stuff on this page about eigenvalues. Is the rank definition correct? —Preceding unsigned comment added by Yahastu ( talk • contribs) 15:08, 20 January 2009 (UTC)
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)
I think the article would be greatly improved if there was a small section after the Definition called Examples showing a simple example of a 2x2 matrix which is positive-definate and an example of a 2x2 matrix which is not. For the latter it would be great to show a single vector which proves this fact. —Preceding unsigned comment added by Pupdike ( talk • contribs) 17:43, 21 March 2009 (UTC)
I went ahead and added the example section. I think it serves to improve the value of the page, but please let me know if you feel otherwise. Pupdike ( talk) 22:08, 23 March 2009 (UTC)
An example of positive definite matrix
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) —Preceding unsigned comment added by Kjetil1001 ( talk • contribs)
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)
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)
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)
(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))
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)
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)
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)
That'd be my question too, more or less. In the section "Quadratic forms" it is said: "It turns out that the matrix M is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function." I would be really surprised that symmetric is a necessary condition for positive definite. In fact, PAR gave the counter example, the symmetric part of [2 2; 0 1] is positive definite as can be seen from its eigenvalues, and so is the matrix itself. But I don't know what the author of that section had in mind... — Preceding
unsigned comment added by
137.226.57.179 (
talk)
09:13, 14 June 2013 (UTC)
This is just a question of convention. Any real square matrix can be written as a sum of a symmetric matrix, , and an antisymmetric matrix, . Then we have the dot product . Since dot products are symmetric, the term involving the antisymmetric matrix always vanishes: , so , which means it must be zero. Because of this, many people do not bother to define positive-definiteness for non-symmetric matrices. In fact, restricting positive-definite to apply only to symmetric matrices means that we can say that a matrix is positive-definite if and only if all its eigenvalues are positive. This statement would not be true if positive-definite matrices were allowed to be non-symmetric. But again, in the end, this is just a question of definition, and the definition in which positive-definiteness implies symmetry seems to be the more common one.
Legendre17 (
talk)
19:25, 12 August 2013 (UTC)
Under characterizations, the eigendecomposition is wrong. I think its supposed to be PDP^-1. If you follow the link "unitary matrix" it says the same (U = PDP* with P* = P^-1). Thus the proof is also wrong (maybe) — Preceding unsigned comment added by 188.155.117.79 ( talk) 21:24, 25 April 2014 (UTC)
I want to see a proof for that property, especially the part that is symmetric.
The symmetric matrix
has the required from (with m(0)=1, m(1)=-0.8, m(2)=-0.1) and satisfies
but is not positive definite. 141.34.29.108 ( talk) 21:34, 6 February 2015 (UTC)
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)
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)
In the article I read under Simultaneous diagonalization I read the statement: "". I am almost certain that the following is meant: "". Can someone well-versed in matrix algebra confirm this? Redav ( talk) 18:12, 27 January 2019 (UTC)
The page contains the following statement:
"More generally, a twice-differentiable real function on real variables has local minimum at arguments if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices."
However, positive semi-definite Hessian doesn't guarantee local minimum, it may also be a saddle point. So either it must be changed to "positive definite", or it may be stated that if a point is local minimum then Hessian must be positive semi-definite.
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)
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)
D.Lazard recently reverted my edits on the intro section which, among other things, defined a positive definite matrix in terms of the angle between and . D.Lazard pointed out that this only applies in Euclidean spaces, so strictly speaking the definition should only be made in terms of inner products. I recognize this point and thank D.Lazard for the correction.
However, I still think that the intro section could use some clean up and clarification and that the rest of my edits were essentially good. I'd also like to point out that the current intro section still has a sentence which asserts that the angle between and is "within an angle of of ," without the clarification that this only applies in Euclidean space.
There are a couple of issues here. First, if we're going to stick with the general outline of the current intro, where positive definite, positive semi-definite, negative definite, and negative semi-definite are all defined separately, then I think it would make sense for the very first sentence to go something like this:
I think it's important to include the connection to right angles in Euclidean space because, at least going from my own experience, the whole concept of (positive) definite matrices only started to make sense to me when I read an explanation that made that connection. At bare minimum, we should point out that is an inner product between and — while that may be kind of obvious to anyone who already understands these concepts, to someone who is new to linear algebra it just looks like a rather arbitrary sequence of matrix-vector multiplications. Since the intro already has a sentence explaining that means the transpose of - a sentence which honestly seems a little unnecessary for me- it seems like we are already on some level committed to making this article accessible for beginners in linear algebra.
Secondly, I think there's an argument to be made that since the title of the article is Definite matrix and not Positive definite matrix, the first sentence should actually be some kind of more general definition of definite matrices, and not a definition of positive definite ones. It might be tricky to come up with an easy-to-understand definition of the general concept before explaining the specific categories, though, so I'd only support this change if we can come up with a really clear definition that flows well into the definitions of positive definite, positive semi-definite, etc. Montgolfière ( talk) 19:45, 4 May 2021 (UTC)
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)
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)
Some of the equivalent conditions in the first section are incorrect. The matrix does not have to be symmetric. See for example https://math.stackexchange.com/questions/1954167/do-positive-semidefinite-matrices-have-to-be-symmetric 158.38.1.98 ( talk) 10:59, 23 May 2024 (UTC)
Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.It seems that you consider these generalizations as a norm and that this is incorrect to not emphasizing them. This is a legitimate opinion, but other editors, including myself, have an opposite opinion. In any case, there is nothing incorrect in the present state of the lead. D.Lazard ( talk) 12:54, 23 May 2024 (UTC)