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%md <H1>Row Reduction</H1>

Row Reduction

Here is a matrix over the rational numbers and its reduced echelon form.

C = Matrix(QQ,[[1,1,1,1],[5,1,2,2]]);C
print '--------------------'
C.echelon_form()
[1 1 1 1] [5 1 2 2] -------------------- [ 1 0 1/4 1/4] [ 0 1 3/4 3/4]

Here is a matrix over the integers mod 2 and its reduced echelon form.

H=Matrix(Integers(2),[[1,1,0,0,1,1],[1,0,1,0,1,0],[1,0,1,1,0,1],[0,1,1,1,0,0]]);H
print '--------------------'
H.echelon_form()
[1 1 0 0 1 1] [1 0 1 0 1 0] [1 0 1 1 0 1] [0 1 1 1 0 0] -------------------- [1 0 1 0 0 0] [0 1 1 0 0 1] [0 0 0 1 0 1] [0 0 0 0 1 0]

H=Matrix(Integers(2),[[1,1,0,1,0,0,1],[1,0,1,0,1,0,0],[0,1,1,0,0,1,0]]);H
print '*****************'
H.echelon_form()
***************** [1 0 1 0 1 0 0] [0 1 1 0 0 1 0] [0 0 0 1 1 1 1]


H2=Matrix(Integers(2),[[1,1,0,1,1,1],[1,0,1,1,0,0],[0,1,1,0,1,0]]);H2
print '*****************'
H2.echelon_form()
[1 1 0 1 1 1] [1 0 1 1 0 0] [0 1 1 0 1 0] ***************** [1 0 1 1 0 0] [0 1 1 0 1 0] [0 0 0 0 0 1]



A=[[2,1,0,4],[-1,2,1,0],[0,-1,2,1]]
A
[1 1 0 1 0 0] [1 0 1 0 1 0] [0 1 1 0 0 1] [[2, 1, 0, 4], [-1, 2, 1, 0], [0, -1, 2, 1]]
matrix(A)
[ 2 1 0 4] [-1 2 1 0] [ 0 -1 2 1]
Edit text...
var (' D, P')
A = Matrix (QQ, [[4, 1, 0], [1, 5, 1], [0, 1, 4]]);A

(D, P) [4 1 0] [1 5 1] [0 1 4]
(D,P)=A.right_eigenmatrix();(D,P)
([6 0 0] [0 4 0] [0 0 3], [ 1 1 1] [ 2 0 -1] [ 1 -1 1])
P.inverse()*A*P

[6 0 0] [0 4 0] [0 0 3]
A2=Matrix(QQ,[[8,1,0],[1,5,1],[0,1,7]])
P=A2.right_eigenmatrix()[1]
P.numerical_approx(digits=3)
print '------------------'
D=(P.inverse()*A2*P);D.numerical_approx(digits=3)
[ 1.00 1.00 1.00] [ -3.65 -0.726 0.377] [ 1.38 -2.65 0.274] ------------------ [ 4.35 0.000 0.000] [0.000 7.27 0.000] [0.000 0.000 8.38]
 A=Matrix(QQ,[[1, 0, 0],[0,2,1],[1,-1,4]])
A.right_eigenmatrix()

([1 0 0] [0 3 0] [0 0 3], [ 1 0 0] [ 1/4 1 0] [-1/4 1 0])
A=Matrix(QQ,[[2,1],[2,3]])
A.exp()

[1/3*e^4 + 2/3*e 1/3*e^4 - 1/3*e] [2/3*e^4 - 2/3*e 2/3*e^4 + 1/3*e]
Matrix?
File: /ext/sage/sage-8.1/src/sage/matrix/constructor.pyx
Signature : Matrix(self, ring=None, nrows=None, ncols=None, sparse=None, *args)
Docstring :
Create a matrix.

This implements the "matrix" constructor:

   sage: matrix([[1,2],[3,4]])
   [1 2]
   [3 4]

It also contains methods to create special types of matrices, see
"matrix.[tab]" for more options. For example:

   sage: matrix.identity(2)
   [1 0]
   [0 1]

INPUT:

The matrix command takes the entries of a matrix, optionally
preceded by a ring and the dimensions of the matrix, and returns a
matrix.

The entries of a matrix can be specified as a flat list of
elements, a list of lists (i.e., a list of rows), a list of Sage
vectors, a callable object, or a dictionary having positions as
keys and matrix entries as values (see the examples). If you pass
in a callable object, then you must specify the number of rows and
columns. You can create a matrix of zeros by passing an empty list
or the integer zero for the entries.  To construct a multiple of
the identity (cI), you can specify square dimensions and pass in c.
Calling matrix() with a Sage object may return something that makes
sense. Calling matrix() with a NumPy array will convert the array
to a matrix.

The ring, number of rows, and number of columns of the matrix can
be specified by setting the "ring", "nrows", or "ncols" keyword
parameters or by passing them as the first arguments to the
function in specified order. The ring defaults to "ZZ" if it is not
specified and cannot be determined from the entries. If the number
of rows and columns are not specified and cannot be determined,
then an empty 0x0 matrix is returned.

INPUT:

* "ring" -- the base ring for the entries of the matrix.

* "nrows" -- the number of rows in the matrix.

* "ncols" -- the number of columns in the matrix.

* "sparse" -- create a sparse matrix. This defaults to "True"
  when the entries are given as a dictionary, otherwise defaults to
  "False".

* "entries" -- see examples below.

OUTPUT:

a matrix

EXAMPLES:

   sage: m = matrix(2); m; m.parent()
   [0 0]
   [0 0]
   Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

   sage: m = matrix(2,3); m; m.parent()
   [0 0 0]
   [0 0 0]
   Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

   sage: m = matrix(QQ,[[1,2,3],[4,5,6]]); m; m.parent()
   [1 2 3]
   [4 5 6]
   Full MatrixSpace of 2 by 3 dense matrices over Rational Field

   sage: m = matrix(QQ, 3, 3, lambda i, j: i+j); m
   [0 1 2]
   [1 2 3]
   [2 3 4]
   sage: m = matrix(3, lambda i,j: i-j); m
   [ 0 -1 -2]
   [ 1  0 -1]
   [ 2  1  0]

   sage: matrix(QQ, 2, 3, lambda x, y: x+y)
   [0 1 2]
   [1 2 3]
   sage: matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1))
   [  1 1/2 1/3 1/4 1/5]
   [  2   1 2/3 1/2 2/5]
   [  3 3/2   1 3/4 3/5]
   [  4   2 4/3   1 4/5]
   [  5 5/2 5/3 5/4   1]

   sage: v1=vector((1,2,3))
   sage: v2=vector((4,5,6))
   sage: m = matrix([v1,v2]); m; m.parent()
   [1 2 3]
   [4 5 6]
   Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

   sage: m = matrix(QQ,2,[1,2,3,4,5,6]); m; m.parent()
   [1 2 3]
   [4 5 6]
   Full MatrixSpace of 2 by 3 dense matrices over Rational Field

   sage: m = matrix(QQ,2,3,[1,2,3,4,5,6]); m; m.parent()
   [1 2 3]
   [4 5 6]
   Full MatrixSpace of 2 by 3 dense matrices over Rational Field

   sage: m = matrix({(0,1): 2, (1,1):2/5}); m; m.parent()
   [  0   2]
   [  0 2/5]
   Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

   sage: m = matrix(QQ,2,3,{(1,1): 2}); m; m.parent()
   [0 0 0]
   [0 2 0]
   Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

   sage: import numpy
   sage: n = numpy.array([[1,2],[3,4]],float)
   sage: m = matrix(n); m; m.parent()
   [1.0 2.0]
   [3.0 4.0]
   Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

   sage: v = vector(ZZ, [1, 10, 100])
   sage: m = matrix(v); m; m.parent()
   [  1  10 100]
   Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
   sage: m = matrix(GF(7), v); m; m.parent()
   [1 3 2]
   Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7

   sage: g = graphs.PetersenGraph()
   sage: m = matrix(g); m; m.parent()
   [0 1 0 0 1 1 0 0 0 0]
   [1 0 1 0 0 0 1 0 0 0]
   [0 1 0 1 0 0 0 1 0 0]
   [0 0 1 0 1 0 0 0 1 0]
   [1 0 0 1 0 0 0 0 0 1]
   [1 0 0 0 0 0 0 1 1 0]
   [0 1 0 0 0 0 0 0 1 1]
   [0 0 1 0 0 1 0 0 0 1]
   [0 0 0 1 0 1 1 0 0 0]
   [0 0 0 0 1 0 1 1 0 0]
   Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

   sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent()
   Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

   sage: R = PolynomialRing(QQ, 9, 'x')
   sage: A = matrix(R, 3, 3, R.gens()); A
   [x0 x1 x2]
   [x3 x4 x5]
   [x6 x7 x8]
   sage: det(A)
   -x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

AUTHORS:

* William Stein: Initial implementation

* Jason Grout (2008-03): almost a complete rewrite, with bits and
  pieces from the original implementation

* Jeroen Demeyer (2016-02-05): major clean up, see
  https://trac.sagemath.org/20015 and
  https://trac.sagemath.org/20016