Spectral theorum

>>8036202
nigga no need for no big shot
1/fix an epsilon
2/start with the classical algorithm to find the biggest eigenvalue and an associated eigenvector (power method or a variation of the power method)
once you have that, you see that you only need to scale the eigenvector v1 you found so that ||Av1 - a1 v1||<epsilon.

Now you can just repeat by changing the matrix A into a matrix restricted to a subspace that excludes v1.

Eventually you get all the eigenvalues you want, the associated eigenvectors and the transformation matrix.

This gives priority to the highest eigenvalues (in magnitude). IMO this makes sense since they contain most of the information. But feel free to modify this to match your expectations


Is this OK or did I misunderstand something?

>>8036352
>to find the biggest eigenvalue
You immediately pooped your pants here.

>>8036202
Seems hard to believe. A can be any matrix? Not just hermetian or normal? Are the v_i supposed to be nearly orthogonal?

Seems like you are saying all matrices are nearly diagonalizable.

ADDENDUM: MATRIX IS HERMITIAN

>>8036816
Sorry for that. It's hermitian.

>>8036772
how? Because you're a mathfag and don't know you can perturbate matrices?

The post was just the gist of it. I won't attach a fucking functional code that works in all cases.

OP can adapt this to whatever he needs to do.

>>8036870
If it's Hermetian then it can be diagonalized exactly.

https://en.wikipedia.org/wiki/Hermitian_matrix#Properties

there is no need for approximations.

>>8037729
Bullshit. Read the question.

>>8038276
Which part is bullshit? Hermetian matrices are diagonalizable by a unitary matrix. The question is either has an obvious answer or poorly stated.

>>8036202
try to do the finite case first, like the finite matrices acting on the complex vector space C^n

>>8040132
That's exactly the question

>>8040727
BUMP FOR WHAT

>>8039681
What do with degenerate eigenvalues?

Is this from Rudin's Functional Analysis? Looks like the print.

>>8041276
>"degenerate"

>>>/pol/

do you guys have some kind of autism? what the fuck are you talking about? what's a "computable/constructive" spectral theorem?

>>8041276

It's right there in Wikipedia...

"The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. Moreover, Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of C^n consisting of n eigenvectors of A."

It always works.

>>8042138
Just fuck you, piece of an idiot.

>>8041378
I think OP is not right in the head.

>>8042519
You think Wikipedia is wrong, or what?

>>8043797
Leave the thread

>>8044689
If I leave you will be alone. You are asking for the solution to a problem that is well known and solved.

an entry on the nlab discussing
https://ncatlab.org/nlab/show/computable+physics

>>8036202
The book
Harold M. Edwards-Essays in constructive mathematics-Springer (2004) has a section on the spectral theorem and a detailed proof of existence. He claims that the explicit formulations of the spectral matrices is through another algorithm:

>Therefore, the spectral decomposition of S can be given once the Moore-Penrose generalized inverses of the matrices I — S + pil are found. Note that the computation of the Moore-Penrose generalized inverse of a matrix requires exact computations with the entries, so
it becomes possible only after a splitting field for the minimum polynomial is constructed; the interpretation of the p's and P's as real numbers requires an identification of the splitting field with a subfield of the field of real numbers.

Does it answer your question?

>>8045582
No. Look at the statement. The matrix entries are integers. Furthermore, I don't need an exact decomposition.

Op, there is an important article by Bridges about the uniqueness of the measure. They discuss in greater detail the work by bishop:
Absolute continuity and the uniqueness of the constructive functional calculus.
http://libgen.io/scimag/ads.php?doi=10.1002%2Fmalq.19940400408&downloadname=
The thesis by Havea has a section on the spectral theorem
http://www.ir.canterbury.ac.nz/bitstream/handle/10092/5610/dediu_thesis.pdf?sequence=1&isAllowed=y

So again, if they do not care, you must do the work yourself, at leas tin finite dimension. Real well Chapter 6 and chapter 8 I guess.

File: 1451548804248.png (152KB, 1180x1135px) Image search: [Google]

152KB, 1180x1135px

I saw that Paul Taylor himself answered you on about the reals. Nice.

Anyway, I just come here to recall, to anybody, that Henri lombardi published his book about constructive algebra (on libgen) which is not about linear algebra.
http://hlombardi.free.fr/publis/LivresBrochures.html

>>8046857
Well, it seems the conclusion can be made. This turns out to be an open problem as for now. But addressing it might be not a cakewalk. Perhaps, I should work on it.

>>8046885
From I understood, the reals numbers used in ASD have bad behavior in that there is no computable bound on iterations that allows you to compute a given number up to a predefined precision.

I personally feel much better with the usual Cauchy real numbers.