Intro Document.ipynb

Kernel: SageMath 10.1

## Hello All!  This is a place we can work on our shared Linear Algebra Tutorial.  What should we put in this document?

## Here is an existing Sage Tutorial in html format. We can use it as a guide to get started: https://doc.sagemath.org/html/en/tutorial/tour_linalg.html

## Here is another example that might be of use:

## https://sites.math.washington.edu//~billey/colombia/sage.code/Algebraic%20Combinatorics%20Tutorial.pdf

A = Matrix(QQ, [[1,2,3],[3,2,1],[1,1,1]])

A

[1.00000000000000 + 1.00000000000000*I                      2.00000000000000                      3.00000000000000]
[                     3.00000000000000                      2.00000000000000                      1.00000000000000]
[                     1.00000000000000                      1.00000000000000                      1.00000000000000]

A*A

[10  9  8]
[10 11 12]
[ 5  5  5]

A.echelon_form()

[ 1  0 -1]
[ 0  1  2]
[ 0  0  0]

A.det()

0

A.charpoly()

x^3 - 4*x^2 - 5*x

A.eigenvalues()

[5, 0, -1]

A.is_diagonalizable()

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In [15], line 1
----> 1 A.is_diagonalizable()
File /ext/sage/10.1/src/sage/matrix/matrix2.pyx:11699, in sage.matrix.matrix2.Matrix.is_diagonalizable()
  11697     raise ValueError('matrix entries must be from a field')
  11698 if not A.base_ring().is_exact():
> 11699     raise ValueError('base field must be exact, but {} is not'.format(A.base_ring()))
  11700 
  11701 # check if the sum of algebraic multiplicities equals to the number of rows
ValueError: base field must be exact, but Complex Field with 53 bits of precision is not

A.

A = Matrix(QQ, [[1,2,3],[3,2,1],[1,1,1]])

A.diagonalization()

(
[ 5  0  0]  [    1     1     1]
[ 0  0  0]  [13/11    -2    -1]
[ 0  0 -1], [ 6/11     1     0]
)

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [21], line 1
----> 1 B = Matrix(_ ,[[Integer(1),Integer(2),Integer(3)],[Integer(3),Integer(2),Integer(1)],[Integer(1),Integer(1),Integer(1)]])
File /ext/sage/10.1/src/sage/matrix/constructor.pyx:643, in sage.matrix.constructor.matrix()
    641 """
    642 immutable = kwds.pop('immutable', False)
--> 643 M = MatrixArgs(*args, **kwds).matrix()
    644 if immutable:
    645     M.set_immutable()
File /ext/sage/10.1/src/sage/matrix/args.pyx:356, in sage.matrix.args.MatrixArgs.__init__()
    354         if argi == argc: return
    355 
--> 356     raise TypeError("too many arguments in matrix constructor")
    357 
    358 def __repr__(self):
TypeError: too many arguments in matrix constructor