Talk:Singular value

The definition of singular value given here is not general enough for the article singular value decomposition, since over there we need the concept for non-square matrices as well. AxelBoldt 16:41, 20 Apr 2005 (UTC)

Why "singular"?
What's "singular" about a singular value? (Why the name?) —Ben FrantzDale 16:56, 20 November 2006 (UTC)
 * According to my math prof, it's just that these are the values that make $$(\sqrt{A^*A}-\lambda I)$$ singular. —Ben FrantzDale 21:02, 5 December 2006 (UTC)

What is |A|?
What is the meaning of |A| as mentioned in the article? --Drizzd (talk) 09:16, 4 April 2008 (UTC)


 * I am told that |A| is in fact defined as $$\sqrt{A^* A}$$, and therefore the previous formulation did not make any sense. I now expanded |A| to $$U |\Lambda| U^*$$ for the unitary diagonalization $$A = U \Lambda U*$$. $$|\Lambda|$$ can now be interpreted as either $$\sqrt{\Lambda* \Lambda}$$ or as taking element-wise absolute value, because both give the same result. --Drizzd (talk) 20:37, 4 April 2008 (UTC)

S-number
Who the hell calls them "s-number"? That's the first time I've heard the term. I'm moving this back to "Singular value". Swap (talk) 22:31, 3 May 2008 (UTC)

that s the first time i heard that too مبتدئ (talk) 17:27, 4 November 2008 (UTC)

I would like to second that. If there is no reference where singular values are called s-numbers, it should be removed. 70.92.123.71 (talk) 17:46, 12 October 2019 (UTC)

singular value and operator norm
I have a question: It is known that the max singular value is an operator norm (more precisly the one described here talks about the through the ||. ||2 induced operator Norm). Since the max singular value is a kind of strongest amplification of a system/ norm and since one can use different norms to define an induced operator norm, is the max singular value only limited or defined with the 2 induced norm or can one define it with any other norm???? Best regards مبتدئ (talk) 17:27, 4 November 2008 (UTC)

I mean max singular value is defined as:

$$max_{|u|_{2}}\frac{||Au||_{2}}{||u||_{2}}$$

with $$||u||_{2}= \sum u_{i}^{2}$$

now if i take another norm different from the 2 Norm, does this define a max singular value too??? thanks مبتدئ (talk) 17:37, 4 November 2008 (UTC)

other formulation of the question: the max singular value is the induced 2 norm of the operator. Is the induced infinity or the induced 5 norm of the operator also a/ the max singular value ??? مبتدئ (talk) 17:51, 4 November 2008 (UTC)

Chapter needed
Can anyone add a chapter about what a structured singular value is? مبتدئ (talk) 01:40, 17 November 2008 (UTC)

Merge with Singular Value Decomposition
Wouldn't it be better to merge this article with Singular Value Decomposition? It's far more complete and defines singular values in a much better way then this article does... anoko_moonlight (talk) 12:32, 20 January 2009 (UTC)

HIstory
The history parts of this article do not agree with the Wikipedia entry Singular value decomposition. There we read that the term singular value was coined by Picard, while here the credit goes to Smithies. Who is right?

In linear algebra... (proposal for new text to be inserted in the article)
In Linear Algebra, an eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector x so that

 $$      Ax=\lambda x   $$

A singular value and pair of singular vectors of a square or rectangular matrix A are a non-negative scalar σ and two nonzero vectors u and v so that

' $$     Av = \sigma u $$'' $$     A^H u = \sigma v $$'''

The superscript on $$A^H$$ stands for Hermitian transpose and denotes the complex conjugate transpose of a complex matrix. If the matrix is real, then $$A^T$$ denotes the same matrix.

78.38.243.139 (21:15, 7 August 2011)

s-number: redirect
Hello,

singular values are also called s-numbers as stated in the article, but s-number redirects to Meter Point Administration Number without any notice that singular value is also a valid meaning. Can someone with knowledge in the English Wikipedia (I only edit the German one regulary) add this?

Kind regards,

ThE cRaCkEr 14:25, 28 March 2012 (UTC) — Preceding unsigned comment added by ThE cRaCkEr (talk • contribs)

"If T is self-adjoint, then the largest singular value s1(T) is equal to the operator norm of T"
Why does T have to be self-adjoint for max singular value to become operator norm?

$$max_x \frac{||Tx||}{||x||} = max_x \frac{||UDV^*x||}{||x||} = max_y \frac{||Dy||}{||y||} = max s(T)$$, where $$y = V^* x$$


 * You are right, removed the self-adjoint condition because it is unnecessary and really confusing. Yvanko55 (talk) 19:06, 3 August 2016 (UTC)

In complex dynamics
There is also a notion of singular value in complex dynamics; namely for an entire holomorphic function f:\CC\to\CC, w\in\CC is a singular value iff it is a branch point of f as a covering, that is, for any neighbourhood V\in\CC of w, there is a component U_i of f^{-1}(V) so f is not injective on U_i. This notion should also be mentioned somewhere ...