sympy:matrices

from sympy.printing.latex import print_latex from sympy.matrices import * import numpy as np

def ltx(x): return print_latex(x, mat_str='bmatrix',mat_delim=None )

# http://docs.sympy.org/dev/modules/matrices.html

# wikipedia example A = Matrix( np.mat(“1 3 1 4 ; 2 7 3 9 ; 1 5 3 1 ; 1 2 0 8”) )

In [83]: A Out[83]: [1, 3, 1, 4] [0, 1, 1, 1] [0, 2, 2, -3] [0, -1, -1, 4]

In [84]: A.col A.col A.col_del A.col_insert A.col_join A.col_swap A.cols

In [84]: A.nullspace() Out[84]: 2] [-1] [ 1] [ 0

In [85]: ltx(A.nullspace()) \begin{bmatrix}\begin{bmatrix}2\\-1\\1\\0\end{bmatrix}\end{bmatrix}

In [86]: A.ro A.row A.row_del A.row_insert A.row_join A.row_swap A.rows

In [86]: A.rref()[0] Out[86]: [1, 0, -2, 0] [0, 1, 1, 0] [0, 0, 0, 1] [0, 0, 0, 0]

In [87]: ltx(A.rref()[0]) \begin{bmatrix}1 & 0 & -2 & 0\\0 & 1 & 1 & 0\\0 & 0 & 0 & 1\\0 & 0 & 0 & 0\end{bmatrix}

In [88]: print “Colspace”; ltx(A[:,0]); ltx(A[:,1]); ltx(A[:,3]); Colspace \begin{bmatrix}1\\0\\0\\0\end{bmatrix} \begin{bmatrix}3\\1\\2\\-1\end{bmatrix} \begin{bmatrix}4\\1\\-3\\4\end{bmatrix}

In [89]: ltx(A.rref()[0][0:2,:]) \begin{bmatrix}1 & 0 & -2 & 0\\0 & 1 & 1 & 0\end{bmatrix}

In [90]: print “Rowspace ”; ltx(A.rref()[0][0,:]); ltx(A.rref()[0][1,:]); ltx(A.rref()[0][2,:]) Rowspace \begin{bmatrix}1 & 0 & -2 & 0\end{bmatrix} \begin{bmatrix}0 & 1 & 1 & 0\end{bmatrix} \begin{bmatrix}0 & 0 & 0 & 1\end{bmatrix}

In [91]: A. A.C A.col_del A.D A.col_insert A.H A.col_join A.LUdecomposition A.col_swap A.LUdecompositionFF A.cols A.LUdecomposition_Simple A.combine A.LUsolve A.conjugate A.QRdecomposition A.copyin_list A.QRsolve A.copyin_matrix A.T A.cross A.add A.delRowCol A.adjugate A.det A.applyfunc A.det_bareis A.berkowitz A.dot A.berkowitz_charpoly A.eigenvals A.berkowitz_det A.eigenvects A.berkowitz_eigenvals A.evalf A.berkowitz_minors A.expand A.charpoly A.extract A.clone A.eye A.cofactor A.fill A.cofactorMatrix A.get_diag_blocks A.col A.hash A.inv A.row A.inverse_ADJ A.row_del A.inverse_GE A.row_insert A.inverse_LU A.row_join A.is_lower A.row_swap A.is_square A.rows A.is_symbolic A.rref A.is_upper A.shape A.jacobian A.simplify A.key2ij A.slice2bounds A.mat A.submatrix A.minorEntry A.subs A.minorMatrix A.tolist A.multiply A.trace A.multiply_elementwise A.transpose A.norm A.vec A.normalized A.vech A.nullspace A.zero A.permuteBkwd A.zeronm A.permuteFwd A.zeros A.print_nonzero A.project A.reshape

In [92]: A.eigenvals() Out[92]: {0: 1, 1: 1, 2: 1, 5: 1}

In [93]: A.eig A.eigenvals A.eigenvects

In [93]: A.eigenvects() Out[93]: [(1, 1, 1] [0] [0] [0), (2, 1, 2] [ 1),

(0, 1, [[ 2]
[-1]
[ 1]
[ 0]]),
 (5, 1, [[3/4]
 [  0]
 [ -1]
 [  1]])]

 In [94]: ltx( A.eigenvects() )
 \begin{bmatrix}\begin{pmatrix}1, & 1, & \begin{bmatrix}\begin{bmatrix}1\\0\\0\\0\end{bmatrix}\end{bmatrix}\end{pmatrix}, & \begin{pmatrix}2, & 1, & \begin{bmatrix}\begin{bmatrix}9\\\frac{3}{2}\\\frac{1}{2}\\1\end{bmatrix}\end{bmatrix}\end{pmatrix}, & \begin{pmatrix}0, & 1, & \begin{bmatrix}\begin{bmatrix}2\\-1\\1\\0\end{bmatrix}\end{bmatrix}\end{pmatrix}, & \begin{pmatrix}5, & 1, & \begin{bmatrix}\begin{bmatrix}\frac{3}{4}\\0\\-1\\1\end{bmatrix}\end{bmatrix}\end{pmatrix}\end{bmatrix}

ROTATION EXAMPLE from sympy import pi from sympy.matrices import rot_axis3

theta = pi/3 rot_axis3(theta) # [ 1/2, sqrt(3)/2, 0] # [-sqrt(3)/2, 1/2, 0] # [ 0, 0, 1]