Talk:Unitary matrix

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, Mid-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Importance
 Field: Algebra

your a failure at this search stuff

Could the author please replace the good with the bad

—Preceding unsigned comment added by (talk) 16:48, 14 September 2007 (UTC) 

The conjugate transpose is also known as the hermitian adjoint, represented with a dagger. For example:

U^dagger * U = I_n where the carrot represents a superscript and the underscore the subscript.

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

--> Isn't it an orthonormal matrix, rather than just orthogonal?

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)

Just a thought but could someone add in an explaination of the unitary group when the field is finite? Since the definition of the unitary matrix relies on the conjugate transpose is there an equivalent definition of "conjugate transpose on a finite field"? TooMuchMath 20:20, 13 April 2006 (UTC)

"Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose U^* \,." Would it be more precise here to use "if and only if" rather than just "if" ? Or maybe call it an alternative definition? Richard Giuly 05:22, 24 October 2006 (UTC)

Symbol conventions[edit]

This article uses dagger for conjugate transpose, but in conjugate transpose article we use star. Both articles should use the same notation convention. I personally prefer dagger. What are your preferences? Merilius 21:30, 03 March 2008 (UTC)

I prefer both from personal use (quantum mechanics) and for the reduced ambiguity, since * means many things in many contexts. I think, though, that as long as each article uses an internally consistent convention and explains the symbology it is not necessary to regularize them. - Eldereft ~(s)talk~ 21:20, 3 March 2008 (UTC)
Still, it might be useful to explain that is the same as in the context of the conjugate transpose article, as many people will link here from pages such as that and normal matrix Westquote (talk)
Although Conjugate transpose uses start, hermitian matrix uses dagger. All related pages should use the same notation noting that there are other notation in the world. —Preceding unsigned comment added by (talk) 18:50, 24 February 2010 (UTC)

Unimodular Matrices[edit]

Naive question: Both unitary matrices and unimodular matrices have the property |det(U)|=1. Why is there no mention of unitary matrices in the unimodular matrix article and vice versa? I'm curious to know how they're related and under what circumstances matrices can be both unitary and unimodular.

--Error9312 (talk) 20:27, 3 March 2009 (UTC)

A matrix that is both unitary and unimodular is a permutation matrix, with possibly some 1s replaced by -1s. So the two groups have very small intersection.--Roentgenium111 (talk) 20:10, 31 May 2012 (UTC)

matrix dimension versus matrix size[edit]

I have a question about the phrase:

"where In is the identity matrix in n dimensions"

Based on the start of the article it sounds to me like unitary matrices are all 2 dimentional (n x n). Wouldn't it be more correct for the above phrase to be:

"where In is the identity matrix of size n"

Scallop7 (talk) 22:01, 18 June 2009 (UTC)

Examples needed[edit]

Could someone put in some examples of sparse and non-sparce matrices that are unitary? Duoduoduo (talk) 16:10, 25 October 2011 (UTC)

incomplete article and misses major points[edit]

the article's first paragraph states, "note this condition implies that a matrix is unitary if and only if it has an inverse which is equal to its conjugate transpose". Sorry but most people cannot comprehend such implications and neither do they have the ability to "note". Especially if the reader is not a mathematical expert. The article should HOW "it is unitary if and only if it has an inverse which is equal to its conjugate transpose". Instead of using vague terminology like "note", the author of this article needs to do a decent job in explaining. Many people come to wikipedia to learn, not to find missing pieces of puzzle. its written in a manner that assumes the reader already has a math degree. — Preceding unsigned comment added by (talk) 02:53, 7 April 2012 (UTC)

Symbol conventions, redux[edit]

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[edit]

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[edit]

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[edit]

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 ( 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 (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 (talkcontribs) 08:30, 23 December 2015 (UTC)