Add "eigendecomposition" to docstring

[jmantysalo]

It took a while to found commands for doing "eigendecomposition", also called "spectral decomposition", to a matrix. Hence I suggest adding those words to documentation of jordan_form(), so that next one will found them more easy.

Also "Matrix.random(GF(3), 2) never generates [[2,0],[0,2]]" must be changed to have Matrix.random(GF(3), 2), algorithm='unimodular').

matrix2.pyx, rref() has error in link to "echelon_form".

[jmantysalo]

New commits:

​b4b58db

Trivial additions.

[zonova]

I do not believe that the jordon canonical form and the eigen value decomposition are equivalent? I do not know much about this topic, but on this pdf: ​https://www.mathworks.com/moler/eigs.pdf

Section 10.8(page 16) seems to indicate that the JCF(jordon canonical form) gives a numerically imperfect approximation compared to the Schur form? What am I missing?

[jmantysalo]

Replying to zonova:

I do not believe that the jordon canonical form and the eigen value decomposition are equivalent? I do not know much about this topic, but on this pdf: ​https://www.mathworks.com/moler/eigs.pdf

Section 10.8(page 16) seems to indicate that the JCF(jordon canonical form) gives a numerically imperfect approximation compared to the Schur form? What am I missing?

I am not an expert, got this as a comment from a teahcer. Any expert of are here?

Numerical instability is another issue, and does not count if we do not use RR on other inexact base rign.

The page says "If A is not defective, then the JCF is the same as the eigenvalue decomposition." So yes, this should be rephrased.

[mmezzarobba]

Replying to jmantysalo:

I am not an expert, got this as a comment from a teahcer. Any expert of are here?

I'm no expert, but afaik the terms “eigendecomposition” and “spectral decomposition” are used for diagonalizable matrices only, while the Jordan form is defined for all matrices (and coincides with the eigendecomposition when the matrix is diagonalizable).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​10286f8

Eigendecomposition is only defined for diagonalizable matrices.

[jmantysalo]

Replying to mmezzarobba:

Replying to jmantysalo:

(and coincides with the eigendecomposition when the matrix is diagonalizable).

OK, this corrected.

[chapoton]

sage -t --long src/sage/matrix/matrix2.pyx
    Error: TAB character found at lines 9792,9793

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​49b57f9

Tabs to spaces.

[jmantysalo]

Arghs. Emacs and .py vs. .pyx.

[chapoton]

ok, let it be.