Talk:Transpose

message: Another note -- should mention that the generalization of the notion of transpose to complex matrices is to make element (i,j) equal to the conjugate of the (j,i) element. I agree with Chris W that the notation A' should be mentioned as an alternative. Wile E. Heresiarch 08:39, 11 Mar 2004 (UTC)
thread_title:  Note =

message: I was trying to understand what the derivative of a transposed matrix is with respect to that matrix? So something like
 \frac{\mathrm{d}A \Theta}{\mathrm{d}\Theta}
where both are matrices.
yanneman 13:26, 20 November 2006 (UTC)
thread_title:  Differentiation of transposed matrices 

message: In the Transpose of linear maps section, the article had read:
:If f : V→W is a linear map between vector spaces V and W with dual spaces W* and V*, we define the  transpose of f to be the linear map tf : W*→V* with
::
  {}^t f (\phi) = \phi\circ f\,   for every \ \phi in W*.
An anonymous user changed that to
:If f : V→W is a linear map between vector spaces V and W with dual spaces W* and V*, we define the  transpose of f to be the linear map fT : W*→V* with
::
  f (\phi)^{\mathrm{T}} = \phi\circ f\,   for every \ \phi in W*.
Now, I think that's confusing (it's f that's transposed, not f(φ)), and it would be better to write
::
  f^{\mathrm{T}}(\phi) = \phi\circ f\,   for every \ \phi in W*.
But even that seems ambiguous; is this "transpose f" or "f to the T power"? But I'm unfamiliar with this notation, and I certainly don't have the same objection to T used with matrices. So, are there any experts who could weigh in with the most common usage in this area? (I'm not proposing to change the T notation used in the earlier sections, just to keep the t notation in this one section.) --Quuxplusone 21:14, 8 August 2005 (UTC)
replies: text: Putting the T in front sounds like a worthy experiment. In general though, you have to realise that mathematics notation is never anywhere near as rigorous as people would like to think. A classic example is -1 designates both the inverse and the reciprocal. Cesiumfrog (talk) 04:52, 11 May 2010 (UTC)

text: What about using f^\top(\phi)? —Ben FrantzDale (talk) 12:40, 11 May 2010 (UTC)
thread_title:  Transpose of linear maps 

message: hermitian transpose = conjugate transpose? --Moala 09:23, 20 December 2005 (UTC)
thread_title:  hermitian transpose? 

message: In the "Special transpose matrices" section, the writing implies that an orthogonal matrix G is defined as one for which GT=G-1. Thus I was going to change the "if"  in "...that is, G is orthogonal if..." to "iff" but I was unsure if this was really a fundamental definition. The "Orthogonal Matrix" page does the same as this one.
It seems like a decent definition of an orthogonal matrix could be a G such that GGT and GTG are (one or both) diagonal or something. Not necessarily that one, but it's enough to make me suspect there's a more general definition some people might use.
Hopefully someone better versed in (multi-)linear algebra literature comes along and knows if there's a more general definition. If there isn't, or if it's still fully compatible, let's change the "if" to "iff" here and possibly in the "Orthogonal Matrix" page too.  --Horn.imh (talk) 19:20, 16 June 2011 (UTC)
replies: text: I'm pretty sure you are right and that it is iff. Suppose G is not orthogonal. Then two columns of G aren't orthogonal (or a column doesn't have norm of one). Then G^\top G will not be diagonal (because off-diagonal terms are inner products of columns with different columns) in the case that the columns aren't orthogonal, and it will have something other than one on the diagonal in the case that any columns don't have norm of one (because the diagonals are the inner products of columns with themselves). QED. —Ben FrantzDale (talk) 20:05, 16 June 2011 (UTC)
thread_title:  Orthogonal Matrices 

message: o is a terrible notation for anything because it looks like a zero.  We should change this to w or something.  Is the author trying to suggest an \omega?  Because then they should just use the \omega...
Also, we need to be consistent for our transpose notation.  Should it be \textstyle{^t}v \textstyle{^T}v v^T or v^t?  We use three of the four possibilities here.
129.32.11.206 (talk) 19:16, 10 October 2012 (UTC)
thread_title:  Notation in 'Transpose of Linear Maps' 

message: In the section Transpose of linear maps, the abstract definition of a transpose is in principle independent of any bilinear form.  This was stated in this way until changed by this edit (which may have been taken from Linear Algebra Quick Study Guide for Smartphones and Mobile Devices).  This fundamentally changes the definition of a transpose in the abstract context.  It would make more sense to me if it were defined primarily in the metric-free context, and (if desired) related to the concept defined in the section at present when suitable bilinear forms are available.  I suggest reverting this section to the earlier form, with the approach using bilinear forms omitted.  Does anyone with more familiarity of the area know what the most generally accepted definition is?  — Quondum 14:02, 1 June 2013 (UTC)
replies: text: I would call what is described in that section of the article the adjoint rather than the transpose, although I'm not sure whether there is a universally accepted definition.  It would make sense to me to define the transpose in a metric free setting and define the adjoint as a generalization.  I'm a little surprised that we don't already have an article on the adjoint (except for the special case hermitian adjoint.) Sławomir Biały (talk) 14:59, 1 June 2013 (UTC)
replies: text: Thanks.  It is a pity that the definitions seem to be a little variable (gauging from the few references I've browsed).  I'll make a change along these lines in the next week or so, any comment from other editors being welcome.  — Quondum 11:07, 2 June 2013 (UTC)

text: I've made some comprehensive changes to the section, criticism welcome.  I also removed a misguided association of the transpose of a coordinate vector and the more abstract concept of a transpose from Dual basis.  — Quondum 02:00, 5 June 2013 (UTC)
thread_title:  Transpose of linear maps: why defined in terms of a bilinear form? 

message: States \mathbf{A}^{\mathrm{T}} = \mathbf{A}^{*}
but I think it should be \mathbf{A}^{\mathrm{T}} = \overline{\mathbf{A}}. I think this is also what user Moala pointed out below way back in 2005.  — Preceding unsigned comment added by 2601:9:2C80:464:809A:FAF:9F56:4439 (talk) 03:42, 19 May 2014 (UTC)
replies: text: I think this is a matter of notation: sometimes the star is used to denote complex conjugate. I've changed this in the interest of reducing ambiguity and misunderstanding. —Quondum 04:41, 19 May 2014 (UTC)
thread_title: Error in definition of Hermitian matrix?

message: > (also written A′, Atr, tA or At)
Should that be ′ or ʹ? Source? JDAWiseman (talk) 08:18, 9 November 2017 (UTC)
thread_title:  Which Prime character? 

message: The following external link was removed:
* How to create Transpose, yodalearning.com
Clicking on this link calls for enrollment in a course. If someone enrolls and finds good information on Transpose, then it might be used. For now it is just linkspam. — Rgdboer (talk) 22:59, 24 July 2018 (UTC)
thread_title:  Yodalearning 

message: In the text, should "a matrix raised to the Ath power" read instead "a matrix raised to the T-th power"?  — Preceding unsigned comment added by Lehnekbn (talk • contribs) 21:42, 28 January 2019 (UTC)
thread_title:  Matrix to the Ath power ...? 

message: The following was removed:
:The reason transpose of a matrix is used is to get the sin(θ) between two vectors in a matrix. The dot product gives the cos(θ) of two vectors, and if we want to get the sin(θ), we would have to do a cos inverse operation to get the angle or use sin^2 + cos^2 = 1. Transpose of a matrix rotates the matrix angle to its complementary angle of pi/2 changing cos(θ) to sin(θ), allowing us to use simpler equations especially in inverse(A) = transpose(adjoint(A)) / det(A). An equation with sin(θ) and cos(θ) is vastly simpler than one with just cos(θ) to compute all other mathematical equations using tan(θ), tanh(θ), etc.
Perhaps the contributor means adjugate matrix. The business of sin(θ) being found this way is unclear and is given with no reference. — Rgdboer (talk) 22:39, 22 February 2019 (UTC)
thread_title:  Adjugate 

message: In some articles about numerical methods for control systems there is also the notion of the pertransposed matrix.
An example for such an article is: .
The definition there is: "transposed with respect to the main antidiagonal". (Just search for pertransposed within that article.)
If I take this literally for a square matrix A\in\mathbb{R}^{n\times n} I get the pertransposed as A^P = A({\rm end}:-1:1,{\rm end}:-1:1)' with Octave notation.
This is also the way it is defined in .
But, in  it is used as A^P = A({\rm end}:-1:1,{\rm end}:-1:1) without the transposition. Maybe that is just a typo. But, it caused me to search for pertranspose here on Wikipedia.
I needed the pertransposed in the context of the generation of the Quasi-Kronecker form of a matrix pencil (see e.g., Varga's article).
Wouldn't it be great to mention the pertranspose with the definition from articles as one of the generalizations of the transposed matrix here in this article? --TN (talk) 08:14, 8 January 2021 (UTC)
thread_title:  Pertransposed 

message: For the transpose Mt of a real square matrix considered as a linear mapping M : Rn —> Rn, with the standard dot product ⟨v,w⟩, we have the standard fact that  
   ⟨Mv,w⟩ = ⟨v,Mtw⟩ 
for all vectors v, w ∊ Rn.
Yet in the article this fact is buried deep in the article, in the section Adjoint, only in the greatest generality.
The section Adjoint is entirely appropriate. But the basic fact above, in its most common manifestation, ought to be mentioned much, much earlier in the article. Especially because that is often how the transpose is defined. 2601:200:C000:1A0:291B:4FAF:4C47:67FE (talk) 18:18, 24 September 2021 (UTC)
thread_title:  Serious omission 

message: This Wikipedia article can easily overwhelm a beginner. there is no point in giving all the unstructured & unstitched details. The article requires a section on motivation for transposes and the link to least squares and the four fundamental sub spaces. Can we work towards a well directed article? With a couple of volunteers I can take up this responsibility. 103.118.50.5 (talk) 05:46, 18 September 2022 (UTC)
replies: text: Hello Rupnagar. Using this Talk space to start, please explain the 4 fundamental subspaces. And what is the  connection to  least squares? Note that transpose refers to binary relations as well as linear transformations, so various details are required. As for motivation, the relation (mathematics) context is common. Rgdboer (talk) 04:02, 19 September 2022 (UTC)
thread_title:  Overwhelming details without the right intuition