Re: Diagonalization in Minkowski space

On Jul 7, 7:31 am, Eric Gisse <jowr...@xxxxxxxxx> wrote:
On Jul 7, 2:33 am, Imago Mortis <meccanica.quantost...@xxxxxxxxx>
wrote:

Dear Friends

---
As you will soon see, I'm not a native speaking English.
I apologize for my poor language and I hope I'm going
to write almost intelligible sentences.
---

Please, can you point the results (if any) of the theory
of orthogonal diagonalization of matrice, over the real
field endowed with the standard euclidean product, not
extendible to Minkowski space, i.e. depending from the
positive definiteness (positive non degeneracy) of the
product ?

The dot product is not positive definite in Minkowski space.

I don't really know what you are asking.



Best Regard.

Imago Mortis

I think I know what he is asking, although I don't know the
answer. I think this is a serious inquiry, although he has some
difficulty expressing it. This is my best guess as to what he means.
Note that relativity is an advocation, not part of what I do in my
profession. So I tentatively give this interpretation as to his
question, and he can correct me.
Any square matrix that is not "defective" can be diagonalized.
Defective matrices are a mathematical oddity. They have fewer linearly
independent eigenvectors than eigenvalues. The matrices of many
"projection" operators are defective. However, the transformation
operators in relativity are not defective. For example, the matrix for
the Lorentz transformation is not defective, nor is it a "projection
operators."
The eigenvalues and eigenvectors are determined from the matrix in
any one of several possible ways. In other words, find the eigenvalues
c and the eignevectors x for a square matrix M so that
c x=Mx
Then there is a diagonal matrix M' which contains the eigenvalues
c, and a transformation matrix S containing the eigenvectors so that
M'=S^-1 M S.
Note that M does not have to be Hermetian, or even unitary. If M
is not Hermetian, c and X will have complex numbers with imaginary
components.
If by Minkowski metrix matrix he means the metric matrix, then
it is not defective. So the question may be is there any time
diagonalizing the Minkowski matrix is useful. Cite a reference.
You are correct that the inner product of two four vectors is not
positive definite. He may be confused by that point. The inner product
can be positive or negative. However, there is a physical picture that
corresponds to the sign of the inner product. When the inner product
of two "four vector coordinates" that separate two events is positive,
the two events are time-like with respect to each other. When the
inner product of two "four vector coordinates" that separate two
events is negative, the two events are space-like with respect to each
other. So his question may include the qualifier,
"Tell me what diagonalizing the Minkowski metric tells us about
two events that are time-like with respect to each other. Please give
me citations where this problem is treated."
My answer: I have no idea.
.

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