Talk:Unitary matrix

orthonormal vs. orthogonal
A unitary matrix in which all entries are real is an orthogonal matrix.

--> Isn't it an orthonormal matrix, rather than just orthogonal? — Preceding unsigned comment added by Ohthere1 (talk • contribs) 16:11, 17 February 2012 (UTC)


 * In the math world, a real unitary matrix is called an "orthogonal matrix". Maybe that's not the best choice of terms, but that's just the way it is.  I have seen an engineering text use the term "orthonormal matrix".  I don't know if that was the author's personal usage, or if it is common among engineers.  Gsspradlin (talk) 19:21, 25 July 2013 (UTC)

Symbol conventions, redux
The previous discussion about symbol conventions has been inactive for years now, so here is a new discussion about this perhaps least important issue with the article.

The articles conjugate transpose, hermitian adjoint, normal matrix, unitary group, and others, all use the * notation.

If you look at the history of this article, the “H” notation was first used, followed by * and “dagger” in a senseless flip-flop.

In math departments on the US East Coast, so far as I have seen, * notation is standard. Kreyszig; Friedberg, Insel, & Spence; Demmel; and Plato, use *. Gilbert Strang uses "H". Wolfram MathWorld uses "H," as you can see for yourself. However, the explanation they give for not using * is not very strong (especially since it seems to deny any conceptual connection to complex conjugation): "Note that because * is sometimes used to denote the complex conjugate, special care must be taken not to confuse notations from different sources."

Chapter 2 of Applied Functional Analysis, 2E by Oden and Demkowicz offers a decent explanation of dual spaces and adjoints. From where I stand, the bottom line is that * makes sense for conjugate transpose because the conjugate transpose gives a matrix for the Hermitian adjoint of our operator. If our original operator is T from V to W, then the adjoint is a linear operator from W* to V*, the dual spaces. In the case of complex inner product spaces, if we want the duality pairing (.,.) to properly generalize the Hermitian inner product, we take the dual basis to be the antilinear functionals, so that (.,.) is sesquilinear as the inner product is.

Star is well-established for conjugate transpose via the adjoint/dual explanation above. "H" makes a great deal of sense, but has not really caught on so far as I can see. Dagger is ambiguous. In numerical linear algebra, for instance, A^dagger often indicates the Moore-Penrose pseudoinverse. -Undiskedste (talk) 17:57, 26 August 2012 (UTC)

Recent Edits
In addition to the alteration of the notation in the article, I've made a number of edits to remove redundancies and improve readability.

In particular, I have changed the body so that much of what was previously within math tags is now written in the normal way. All such changes were made because there was no reason to have each U stand out from the text; similarly for short equations such as U * = U-1. Other equations remained unchanged.

This article could still use a LOT of improvement, but my interest at the moment was only in making it not as bad as it was before. -Undiskedste (talk) 18:30, 26 August 2012 (UTC)

0x0 matrices
I've corrected the article to change "positive n" to "nonnegative n" when talking about the group of nxn unitary matrices, on the presumption that 0 was excluded simply due to the author's unfamiliarity with degenerate matrices. Hurkyl (talk) 22:55, 7 November 2014 (UTC)


 * Wikipedia understands something different under a degenerate matrix. Do you want to define U(0) as just the trivial group consisting of an empty matrix? I see no point in this, it will just confuse readers if you don't state the case explicitly, and doesn't add anything as all statements on unitary matrices will be trivial for n=0. --Roentgenium111 (talk) 13:26, 22 April 2015 (UTC)

Error in "Elementary constructions" section
It seems to me that there is a mistake in the "general expression of a 2 × 2 unitary matrix" under section "Elementary constructions". To my understanding, Pauli Matrices do not fit into this form, even they are widely recognized to be unitary matrices (https://en.wikipedia.org/wiki/Pauli_matrices). I suspect that the general is restricted to a specific determinant value, but as chemist I prefer do not edit the main article.

Regards, HRS — Preceding unsigned comment added by 163.10.18.83 (talk) 16:52, 17 December 2015 (UTC)


 * The Pauli matrices are a special instance of the general 2x2 expression. Call the elementary 2x2 expression reported in the Wikipedia page by U(a,b,φ). Hence we have: σ1=U(0,1,0), σ2=U(0,-i,0), σ3=U(i,0,-π/2). — Preceding unsigned comment added by Paolostar (talk • contribs) 08:30, 23 December 2015 (UTC)

Determinant angle variable name changes in "2 × 2 unitary matrix" section
Changed angle variable name of first determinant from \theta to \varphi to get determinant consistency in this section.. had to divide-by-two the second existing use of \varphi. Replacing the first use of var \theta with \varphi also gets rid of the double inconsistent use of \theta. TBond (talk) 05:56, 17 June 2017 (UTC)

Incorrect mathematics -- Please check carefully
There are some areas in the article where the mathematics is incorrect. As an example, see the '2x2 unitary matrix' subsection of 'Elementary constructions'. The displayed equation on line 7 of this subsection is incorrect (under the assumption that $$\varphi$$ is such that $$\mathrm{det}(U)=e^{i\varphi}$$). One can very easily see that the determinant of the matrix

$$\left[\begin{matrix}e^{i\varphi_1}\cos\theta & e^{i\varphi_2}\sin\theta \\ -e^{-i\varphi_2}\sin\theta & e^{-i\varphi_1}\cos\theta \end{matrix}\right]$$

is 1. Thus, the matrix

$$U=e^{i\varphi / 2}\left[\begin{matrix}e^{i\varphi_1}\cos\theta & e^{i\varphi_2}\sin\theta \\ -e^{-i\varphi_2}\sin\theta & e^{-i\varphi_1}\cos\theta \end{matrix}\right]$$

has determinant $$e^{i\varphi / 2}$$ and not $$e^{i\varphi}$$. I suspect it's a consequence of multiple edits over time with little care for consistency. I would also like to remind editors that wikipedia is not a place to post 'original research'. Whilst these are elementary exercises and it may be tempting to quickly work something out on paper and type it up in an article, this should never be done in wikipedia for precisely the reason that mistakes such as what I have highlighted can creep in. Whilst citations aren't necessarily expected for every piece of mathematics (as doing so would render the article unreadable), all mathematical material should at least in theory be referenceable to a respectable source such as a textbook.

As a side note, I thoroughly dislike the use of $$\varphi$$ alongside $$\varphi_1$$ and $$\varphi_2$$ on line 7. They are independent variables and it seems completely arbitrary to me that we use two Greek letters $$\varphi$$ and $$\theta$$ to label four independent variables. As it is written, one might be confused as to whether $$\varphi_1$$ and $$\varphi_2$$ bear any relation to $$\varphi$$. --Iteraf (talk) 16:34, 12 July 2018 (UTC)

Errors in "Elementary constructions"??
The referenced (Hartmut-Ziemowit) factorization seems correct, but the text above it seems to be something half-way between SU(2) and U(2)??

Quote from article:

The matrix $U$ can also be written in this alternative form:
 * $$U = e^{i\varphi /2} \begin{bmatrix}

e^{i\varphi_1} \cos \theta & e^{i\varphi_2} \sin \theta \\ -e^{-i\varphi_2} \sin \theta & e^{-i\varphi_1} \cos \theta \\ \end{bmatrix},$$

(end quote)

Just multiplying the determinant, that was identified as $$e^{i\varphi /2}$$ (just above the quote in the article) is where the error is. I THINK?¿?

If it was a factorization of the 2x2 unitary matrix, that is U(2), then this should be true, for some values of $$\alpha, \beta, \xi, \zeta, \phi, \psi,...$$:

$$e^{i\varphi /2}\begin{bmatrix} e^{i\psi} & 0 \\ 0 & e^{-i\psi} \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{bmatrix} \begin{bmatrix} e^{i\Delta} & 0 \\ 0 & e^{-i\Delta} \end{bmatrix} = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{bmatrix} \begin{bmatrix} e^{i\xi} & 0 \\ 0 & e^{i\zeta} \end{bmatrix} \begin{bmatrix} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \\ \end{bmatrix}. $$

But notice that $$\psi, \theta, \Delta$$ are for both columns, while $$\xi, \zeta$$ are only for one. · · · Omnissiahs hierophant (talk) 07:48, 15 May 2021 (UTC)


 * Hello. You undid my removal of the mathematics. Maybe it is like you write, that I was wrong, BUT lets assume that the Hartmut-Ziemowit factorization is correct, then can you please explain to me how
 * $$\underbrace{e^{i\varphi /2}\begin{bmatrix}

e^{i\psi} & 0 \\ 0 & e^{-i\psi} \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{bmatrix} \begin{bmatrix} e^{i\Delta} & 0 \\ 0 & e^{-i\Delta} \end{bmatrix}}_{\text{the contested formula}} = \underbrace{\begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{bmatrix} \begin{bmatrix} e^{i\xi} & 0 \\ 0 & e^{i\zeta} \end{bmatrix} \begin{bmatrix} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \\ \end{bmatrix}}_{\text{Hartmut-Ziemowit}} $$??
 * How do you multiply a matrix that looks something similar to this $$\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$$ or $$\begin{bmatrix} a & 0 \\ 0 & -a \end{bmatrix}$$ (and so on) into something that looks like $$\begin{bmatrix}c & 0 \\ 0 & d\end{bmatrix}$$ (for arbitary values of c, d)?
 * Maybe I missed something obvious...
 * Update - undid my removal, I was likely wrong. · · · Omnissiahs hierophant (talk) 17:47, 24 May 2021 (UTC)

Notation in article clashes with itself
The asterisk is used in this article to denote the conjugate transpose of a matrix. Which is exactly as it should be.

Unfortunately, the asterisk is *also* used in this article to denote complex conjugate.

That is ordinarily a common choice for complex conjugate, but it directly conflicts with the above notation for 1x1 matrices (for which it would mean the conjugate reciprocal complex number).

I don't know much about typography, but surely provides for an overbar to denote complex conjugation?

If all else fails, conj(z) is another common notation for the complex conjugate of z. 2601:200:C000:1A0:ED02:FEE6:7128:44D8 (talk) 03:15, 28 December 2022 (UTC)

The kind of writing that we do not need on Wikipedia
In the section Equivalent conditions both items 4. and 5. use the term "the usual inner product". For instance:

"The columns of U form an orthonormal basis of ℂn with respect to the usual inner product."

If you are too lazy to say what kind of inner product you are talking about or to even provide a link to what you are talking about, it might be better if you waited until you felt more energetic before posting to Wikipedia.