2020-10-07 MATH 3110 Matrix algebra -- Sage

A = matrix(2,2,[1,3,4,2]) B=matrix(2,2,[5,9,10,3]) C=matrix(2,2,[1,2,5,4])

show((A*B)*C)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 125 & 142 \\ 250 & 248 \end{array}\right)$

show(A*(B*C))

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 125 & 142 \\ 250 & 248 \end{array}\right)$

show(A)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 3 \\ 4 & 2 \end{array}\right)$

A.inverse()

[ -1/5  3/10]
[  2/5 -1/10]

[ -1/5  3/10]
[  2/5 -1/10]

A*A.inverse()

[1 0]
[0 1]

[1 0]
[0 1]

A=matrix(3,3,[1,2,3,4,3,2,1,5,6])

B=A.inverse()

A*B

[1 0 0]
[0 1 0]
[0 0 1]

[1 0 0]
[0 1 0]
[0 0 1]

B*A

[1 0 0]
[0 1 0]
[0 0 1]

[1 0 0]
[0 1 0]
[0 0 1]

A=matrix(3,3,[1,2,3,2,3,4,3,4,5])

A.inverse()

Traceback (click to the left of this block for traceback)
...
ZeroDivisionError: Matrix is singular

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_16.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("QS5pbnZlcnNlKCk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmphsNi9n/___code___.py", line 2, in <module>
    exec compile(u'A.inverse()
  File "", line 1, in <module>
    
  File "sage/matrix/matrix2.pyx", line 8503, in sage.matrix.matrix2.Matrix.inverse (/usr/local/sage-6.10/src/build/cythonized/sage/matrix/matrix2.c:63556)
  File "sage/matrix/matrix_integer_dense.pyx", line 3853, in sage.matrix.matrix_integer_dense.Matrix_integer_dense.__invert__ (/usr/local/sage-6.10/src/build/cythonized/sage/matrix/matrix_integer_dense.c:33244)
  File "sage/matrix/matrix_integer_dense.pyx", line 3814, in sage.matrix.matrix_integer_dense.Matrix_integer_dense._invert_flint (/usr/local/sage-6.10/src/build/cythonized/sage/matrix/matrix_integer_dense.c:33078)
ZeroDivisionError: Matrix is singular

A=matrix(3,2,[1,2,4,8,16,32])

A

[ 1  2]
[ 4  8]
[16 32]

[ 1  2]
[ 4  8]
[16 32]

A.transpose()

[ 1  4 16]
[ 2  8 32]

[ 1  4 16]
[ 2  8 32]

A*A.transpose()

[   5   20   80]
[  20   80  320]
[  80  320 1280]

[   5   20   80]
[  20   80  320]
[  80  320 1280]

A.transpose()*A

[ 273  546]
[ 546 1092]

[ 273  546]
[ 546 1092]

(A*A.transpose()).trace()

(A.transpose()*A).trace()

B = matrix(2,3,[5,2,1,7,4,3])

A=B.transpose()*B

show(A)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 74 & 38 & 26 \\ 38 & 20 & 14 \\ 26 & 14 & 10 \end{array}\right)$

show(A.eigenmatrix_right())

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 1.009804864072152? & 0 \\ 0 & 0 & 102.99019513592784? \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 1 \\ -4 & -0.8851861715953374? & 0.518213694531118? \\ 3 & -1.513581562127117? & 0.3576182593748229? \end{array}\right)\right)$

show(A.eigenmatrix_left())

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 1.009804864072152? & 0 \\ 0 & 0 & 102.99019513592784? \end{array}\right), \left(\begin{array}{rrr} 1 & -4 & 3 \\ 1 & -0.8851861715953374? & -1.513581562127117? \\ 1 & 0.518213694531118? & 0.3576182593748229? \end{array}\right)\right)$

show(A.eigenvectors_right())

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(0, \left[\left(1,\,-4,\,3\right)\right], 1\right), \left(1.009804864072152?, \left[\left(1,\,-0.8851861715953374?,\,-1.513581562127117?\right)\right], 1\right), \left(102.99019513592784?, \left[\left(1,\,0.518213694531118?,\,0.3576182593748229?\right)\right], 1\right)\right]$

2020-10-07 MATH 3110 Matrix algebra

953 days ago by calkin