Talk:Spectral theorem

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I think the "Fundamental theorem of Linear Algebra" is normally taken to be the statement that connects kernel, row and column spaces of a matrix. AxelBoldt 02:25 Jan 24, 2003 (UTC)

Added a statement of the spectral theorem for bounded self-adjoint operators. Hopefully this should leave this article in a more-or-less definitive state.CSTAR 21:09, 13 May 2004 (UTC)

Request: The section on the Spectral theorem for unbounded operators is very vague - it more or less just mentions its existence. Since for many applications it is precisely unbounded (differential) operators that are of interest, it would be very useful to at the very least state the Spectral theorem for unbounded self-adjoint operators.

[edit] examples

The section "functional analysis" takes some pains to point out the shift operator, and a scaling operator on L^2[0,1] have no eigenvalues, and then immediately states a theorem that none-the-less, these are unitarily equivalent to some multiplication operator on some measure space. It sure would be nice to have a detailed example for these. linas 05:48, 19 March 2006 (UTC)

[edit] Recent edit

The point is that the sentence

Then—using the fact that Ax = λx iff \overline{A} \overline{x} = \overline{\lambda} \overline{x}

is completely irrelevant. Please remove it.--CSTAR 21:12, 25 April 2006 (UTC)

the following was written immediately prior to the appearance of the above comment:

To CSTAR and any others who are interested: Archelon does not wish to argue about the notation for adjointness of linear operators (or complex|Hermitian conjugation, more accurately) (the notation was changed for the sake of the appearance when rendered, but this[*] is a notoriously volatile issue), but has returned the other things CSTAR removed from the proof that the eigenvalues are real (only some of which were originally added by Archelon, incidentally). Archelon 21:19, 25 April 2006 (UTC)

the following was written immediately thereafter:

The sentence is not irrelevant; it is explanatory. Also, a perusal of the page history will reveal that Archelon is not responsible for it (merely in favour of its retention). Archelon's advice to CSTAR: If you find the sentence intolerable, remove it yourself.

[*] (i.e., the latter) Archelon 21:25, 25 April 2006 (UTC)

Message to archelon (who seems to refer to her/himself in the third person). I did. --CSTAR 21:41, 25 April 2006 (UTC)
Message from Archelon (which does indeed refer to itself in the third person). As you like. Archelon 22:55, 25 April 2006 (UTC)

[edit] Proof

Can anyone explain this line, it wasn't entirely clear for me:

This is finite-dimensional, and A has the property that it maps every vector w in K into K

How do we know that anything in K is mapped into K? Why can't it be mapped into Span(e)? Can anyone clarify, please? 216.7.201.43 13:47, 10 August 2006 (UTC)

Is that paragraph clearer now? --CSTAR 14:59, 10 August 2006 (UTC)
Perfect, that makes sense now. It was hard to follow before. 216.7.201.43 15:32, 10 August 2006 (UTC)
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