Add "eigendecomposition" to docstring

[f29946bc-ee7b-48cd-9abc-3445948c551d]

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".

CC: @rbeezer @kedlaya

Component: documentation

Author: Jori Mäntysalo

Branch/Commit: 49b57f9

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/20465

[f29946bc-ee7b-48cd-9abc-3445948c551d]

Description changed:

--- 
+++ 
@@ -1,2 +1,3 @@
 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')`.

[f29946bc-ee7b-48cd-9abc-3445948c551d]

Description changed:

--- 
+++ 
@@ -1,3 +1,5 @@
 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".

[f29946bc-ee7b-48cd-9abc-3445948c551d]

Branch: u/jmantysalo/addeigendecompositionto_docstring

[f29946bc-ee7b-48cd-9abc-3445948c551d]

Author: Jori Mäntysalo

[f29946bc-ee7b-48cd-9abc-3445948c551d]

New commits:

b4b58db

Trivial additions.

[f29946bc-ee7b-48cd-9abc-3445948c551d]

Commit: b4b58db

[c370f75d-aa64-4088-b68b-2377dcc777c5]

comment:5

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?

[f29946bc-ee7b-48cd-9abc-3445948c551d]

comment:6

Replying to @sagetrac-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.

[mezzarobba]

comment:7

Replying to @jm58660:

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).

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

10286f8

Eigendecomposition is only defined for diagonalizable matrices.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from b4b58db to 10286f8

[f29946bc-ee7b-48cd-9abc-3445948c551d]

comment:9

Replying to @mezzarobba:

Replying to @jm58660:

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

OK, this corrected.

[fchapoton]

comment:10

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 10286f8 to 49b57f9

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

49b57f9

Tabs to spaces.

[f29946bc-ee7b-48cd-9abc-3445948c551d]

comment:12

Arghs. Emacs and .py vs. .pyx.

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:13

ok, let it be.

[vbraun]

Changed branch from u/jmantysalo/addeigendecompositionto_docstring to 49b57f9