def reflect_x():
'''
Inpute: None.
Output: 3x3 X-reflection matrix.
'''
M = identity()
M[('y','y')] = -1
return M
def reflect_y():
'''
Input: None.
Output: 3x3 Y-reflection matrix.
'''
M = identity()
M[('x','x')] = -1
return M
def reflect_y():
'''
Input: None.
Output: 3x3 Y-reflection matrix.
'''
M = identity()
M[('x', 'x')] = -1
return M
def reflect_x():
'''
Inpute: None.
Output: 3x3 X-reflection matrix.
'''
M = identity()
M[('y', 'y')] = -1
return M
def translation(x,y):
'''
Input: An x and y value by which to translate an image.
Output: Corresponding 3x3 translation matrix.
'''
M = identity()
M[('x','u')] = x
M[('y','u')] = y
return M
def scale(a, b):
'''
Input: Scaling parameters for the x and y direction.
Output: Corresponding 3x3 scaling matrix.
'''
M = identity()
M[('x','x')] = a
M[('y','y')] = b
return M
def scale(a, b):
'''
Input: Scaling parameters for the x and y direction.
Output: Corresponding 3x3 scaling matrix.
'''
M = identity()
M[('x', 'x')] = a
M[('y', 'y')] = b
return M
def translation(x, y):
'''
Input: An x and y value by which to translate an image.
Output: Corresponding 3x3 translation matrix.
'''
M = identity()
M[('x', 'u')] = x
M[('y', 'u')] = y
return M
def scale_color(scale_r, scale_g, scale_b):
'''
Input: 3 scaling parameters for the colors of the image.
Output: Corresponding 3x3 color scaling matrix.
'''
M = identity({'r', 'g', 'b'})
M[('r', 'r')] = scale_r
M[('g', 'g')] = scale_g
M[('b', 'b')] = scale_b
return M
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
R = mat2rowdict(identity(A.D[0], 1))
return coldict2mat([solve(A, R[i]) for i in range(len(R))])
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
R = mat2rowdict(identity(A.D[0], 1))
return coldict2mat([solve(A, R[i]) for i in range(len(R))])
def scale_color(scale_r,scale_g,scale_b):
'''
Input: 3 scaling parameters for the colors of the image.
Output: Corresponding 3x3 color scaling matrix.
'''
M = identity({'r','g','b'})
M[('r','r')] = scale_r
M[('g','g')] = scale_g
M[('b','b')] = scale_b
return M
def identity(labels = {'x','y','u'}):
'''
In case you have never seen this notation for a parameter before,
the way it works is that identity() now defaults to having labels
equal to {'x','y','u'}. So you should write your procedure as if
it were defined 'def identity(labels):'. However, if you want the labels of
your identity matrix to be {'x','y','u'}, you can just call
identity(). Additionally, if you want {'r','g','b'}, or another set, to be the
labels of your matrix, you can call identity({'r','g','b'}).
'''
return matutil.identity(labels, 1)
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
R = mat2rowdict(identity(A.D[0], one))
return coldict2mat([solve(A, R[i]) for i in range(len(R))])
def find_triangular_matrix_inverse(A): L=[] label_list = [] I = identity(A.D[0],1) R = mat2rowdict(A) w = mat2rowdict(I) label_list.extend(range(len(R))) for k,v in w.items(): print(k,v,) s = triangular_solve(R, label_list, v) L.append(s) return coldict2mat(L)
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
solution = identity(A.D[0],one)
solutionrowdict = mat2rowdict(solution)
return coldict2mat({key:solve(A,vec) for key,vec in solutionrowdict.items()})
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
R = mat2rowdict(identity(A.D[0], one))
return coldict2mat([solve(A, R[i]) for i in range(len(R))])
def identity(labels={'x', 'y', 'u'}):
'''
In case you have never seen this notation for a parameter before,
the way it works is that identity() now defaults to having labels
equal to {'x','y','u'}. So you should write your procedure as if
it were defined 'def identity(labels):'. However, if you want the labels of
your identity matrix to be {'x','y','u'}, you can just call
identity(). Additionally, if you want {'r','g','b'}, or another set, to be the
labels of your matrix, you can call identity({'r','g','b'}).
'''
from matutil import identity
return identity(labels, 1)
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
solution = identity(A.D[0],1)
solutionrowdict = mat2rowdict(solution)
Arowveclist = list(mat2rowdict(A).values())
label_list = list(range(len(Arowveclist)))
return coldict2mat({key:triangular_solve(Arowveclist, label_list, vec) for key,vec in solutionrowdict.items()})
def rotation(angle):
'''
Input: An angle in radians to rotate an image.
Output: Corresponding 3x3 rotation matrix.
Note that the math module is imported.
'''
from math import sin,cos
M = identity()
M[('x','x')] = cos(angle)
M[('y','y')] = cos(angle)
M[('y','x')] = sin(angle)
M[('x','y')] = -sin(angle)
return M
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
from matutil import mat2coldict, coldict2mat, identity
I = identity(A.D[0], 1)
colsDict = mat2coldict(I)
solvedVecs = {i: solve(A, vector) for i, vector in colsDict.items()}
return coldict2mat(solvedVecs)
def rotation(angle):
'''
Input: An angle in radians to rotate an image.
Output: Corresponding 3x3 rotation matrix.
Note that the math module is imported.
'''
from math import sin, cos
M = identity()
M[('x', 'x')] = cos(angle)
M[('y', 'y')] = cos(angle)
M[('y', 'x')] = sin(angle)
M[('x', 'y')] = -sin(angle)
return M
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
from matutil import mat2coldict, coldict2mat, identity
I = identity(A.D[0], 1)
colsDict = mat2coldict(I)
solvedVecs = {i:solve(A,vector) for i,vector in colsDict.items()}
return coldict2mat(solvedVecs)
def find_triangular_matrix_inverse(A):
"""
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
"""
bRows = []
idMat = list(mat2coldict(identity(A.D[0], 1)).values())
for i in idMat:
bRows.append(ts(list(mat2rowdict(A).values()), list(range(len(A.D[0]))), i))
return coldict2mat(bRows)
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
bRows = []
idMat = list(mat2coldict(identity(A.D[0], 1)).values())
for i in idMat:
bRows.append(
ts(list(mat2rowdict(A).values()), list(range(len(A.D[0]))), i))
return coldict2mat(bRows)
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
bRows = []
idMat = list(mat2coldict(identity(A.D[0], one)).values())
for i in idMat:
bRows.append(solve(A, i))
return coldict2mat(bRows)
def find_matrix_inverse(A):
"""
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
"""
bRows = []
idMat = list(mat2coldict(identity(A.D[0], one)).values())
for i in idMat:
bRows.append(solve(A, i))
return coldict2mat(bRows)
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
cols = []
i = identity(A.D[0], 1)
i_cols = mat2coldict(i)
for k in A.D[0]:
cols.append( triangular_solve([ mat2rowdict(A)[k] for k in mat2rowdict(A) ], list(range(len(A.D[0]))), i_cols[k]) )
return coldict2mat(cols)
def rotate_about(x, y, angle):
'''
Input: An x and y coordinate to rotate about, and an angle
in radians to rotate about.
Output: Corresponding 3x3 rotation matrix.
It might be helpful to use procedures you already wrote.
'''
# Auxiliar matrix to translate to the origin
M_translate_to_origin_point = identity()
M_translate_to_origin_point[('x', 'u')] = -x
M_translate_to_origin_point[('y', 'u')] = -y
# Auxiliar matrix to translate to the origin
M_translate_to_original_position = identity()
M_translate_to_original_position[('x', 'u')] = x
M_translate_to_original_position[('y', 'u')] = y
# Rotate the point in the origin
matrix_rotated = rotation(angle) * M_translate_to_origin_point
# Matrix rotated respecto point
matrix_rotated_respect_point = M_translate_to_original_position * matrix_rotated
return matrix_rotated_respect_point
def find_matrix_inverse(A):
'''
Input:
- A: an invertible Mat over GF(2)
Output:
- A Mat that is the inverse of A
Examples:
>>> M1 = Mat(({0,1,2}, {0,1,2}), {(0, 1): one, (1, 0): one, (2, 2): one})
>>> find_matrix_inverse(M1) == Mat(M1.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
>>> M2 = Mat(({0,1,2,3},{0,1,2,3}),{(0,1):one,(1,0):one,(2,2):one})
>>> find_matrix_inverse(M2) == Mat(M2.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
'''
return coldict2mat([solve(A,i) for i in mat2coldict(identity(A.D[0],one)).values()])
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
cols = []
i = identity(A.D[0], one)
i_cols = mat2coldict(i)
for k in A.D[0]:
cols.append( solve(A, i_cols[k]) )
return coldict2mat(cols)
def rotate_about(x,y,angle):
'''
Input: An x and y coordinate to rotate about, and an angle
in radians to rotate about.
Output: Corresponding 3x3 rotation matrix.
It might be helpful to use procedures you already wrote.
'''
# Auxiliar matrix to translate to the origin
M_translate_to_origin_point = identity()
M_translate_to_origin_point[('x','u')] = -x
M_translate_to_origin_point[('y','u')] = -y
# Auxiliar matrix to translate to the origin
M_translate_to_original_position = identity()
M_translate_to_original_position[('x','u')] = x
M_translate_to_original_position[('y','u')] = y
# Rotate the point in the origin
matrix_rotated = rotation(angle)*M_translate_to_origin_point
# Matrix rotated respecto point
matrix_rotated_respect_point = M_translate_to_original_position*matrix_rotated
return matrix_rotated_respect_point
def find_matrix_inverse(A):
'''
input: An invertible matrix, A, over GF(2)
output: Inverse of A
>>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
True
'''
identity_matrix = identity(A.D[0], one)
dict1_colmatrix = mat2coldict(identity_matrix)
newdict1 = {}
for key in dict1_colmatrix:
newdict1[key] = solve(A,dict1_colmatrix[key])
result = coldict2mat(newdict1)
return result
def find_matrix_inverse(A):
'''
Input:
- A: an invertible Mat over GF(2)
Output:
- A Mat that is the inverse of A
Examples:
>>> M1 = Mat(({0,1,2}, {0,1,2}), {(0, 1): one, (1, 0): one, (2, 2): one})
>>> find_matrix_inverse(M1) == Mat(M1.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
'''
coldict = {}
identity_mat = mat2coldict(identity(A.D[0], one))
for col_label in A.D[0]:
coldict[col_label] = solve(A, identity_mat[col_label])
return coldict2mat(coldict)
def find_triangular_matrix_inverse(A):
'''
Supporting GF2 is not required.
Input:
- A: an upper triangular Mat with nonzero diagonal elements
Output:
- Mat that is the inverse of A
Example:
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
aa = [a for a in mat2rowdict(A).values()]
dd = [d for d in A.D[0]]
return coldict2mat([triangular_solve(aa, dd, i) for i in mat2coldict(identity(A.D[0], 1)).values()])
def find_matrix_inverse(A):
'''
Input:
- A: an invertible Mat over GF(2)
Output:
- A Mat that is the inverse of A
Examples:
>>> M1 = Mat(({0,1,2}, {0,1,2}), {(0, 1): one, (1, 0): one, (2, 2): one})
>>> find_matrix_inverse(M1) == Mat(M1.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
>>> M2 = Mat(({0,1,2,3},{0,1,2,3}),{(0,1):one,(1,0):one,(2,2):one})
>>> find_matrix_inverse(M2) == Mat(M2.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
'''
cols = mat2coldict(identity(A.D[0], one))
B = coldict2mat({x:solve(A, cols[x]) for x in cols})
return B
def find_matrix_inverse(A):
'''
Input:
- A: an invertible Mat over GF(2)
Output:
- A Mat that is the inverse of A
Examples:
>>> M1 = Mat(({0,1,2}, {0,1,2}), {(0, 1): one, (1, 0): one, (2, 2): one})
>>> find_matrix_inverse(M1) == Mat(M1.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
>>> M2 = Mat(({0,1,2,3},{0,1,2,3}),{(0,1):one,(1,0):one,(2,2):one})
>>> find_matrix_inverse(M2) == Mat(M2.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
'''
cols = mat2coldict(identity(A.D[0], one))
B = coldict2mat({x: solve(A, cols[x]) for x in cols})
return B
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1):
0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
identity_matrix = identity(A.D[0], one)
dict1_rowmatrix = mat2rowdict(identity_matrix)
newdict1 = {}
for key in dict1_rowmatrix:
newdict1[key] = solve(A,dict1_rowmatrix[key])
result = rowdict2mat(newdict1)
final_result = transpose(result)
return final_result
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
inverseColDict = dict()
rowList = [val for val in mat2rowdict(A).values()]
label_list = [d for d in A.D[0]]
identVecList = [ val for val in mat2coldict(identity(A.D[0] , 1)).values()]
for k in range(len(rowList)):
inverseColDict[k] = triangular_solve(rowList, label_list, identVecList[k])
return coldict2mat(inverseColDict)
def find_triangular_matrix_inverse(A):
'''
Supporting GF2 is not required.
Input:
- A: an upper triangular Mat with nonzero diagonal elements
Output:
- Mat that is the inverse of A
Example:
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
rows = mat2rowdict(A)
cols = mat2coldict(identity(A.D[0], 1))
B = coldict2mat({x:triangular_solve_n(rows, cols[x]) for x in cols})
return B
def find_triangular_matrix_inverse(A):
'''
Supporting GF2 is not required.
Input:
- A: an upper triangular Mat with nonzero diagonal elements
Output:
- Mat that is the inverse of A
Example:
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
rows = mat2rowdict(A)
cols = mat2coldict(identity(A.D[0], 1))
B = coldict2mat({x: triangular_solve_n(rows, cols[x]) for x in cols})
return B
def grayscale():
'''
Input: None
Output: 3x3 greyscale matrix.
'''
M = identity({'r','g','b'})
M[('r','r')] = 77/256
M[('r','g')] = 151/256
M[('r','b')] = 28/256
M[('g','r')] = 77/256
M[('g','g')] = 151/256
M[('g','b')] = 28/256
M[('b','r')] = 77/256
M[('b','g')] = 151/256
M[('b','b')] = 28/256
return M
def grayscale():
'''
Input: None
Output: 3x3 greyscale matrix.
'''
M = identity({'r', 'g', 'b'})
M[('r', 'r')] = 77 / 256
M[('r', 'g')] = 151 / 256
M[('r', 'b')] = 28 / 256
M[('g', 'r')] = 77 / 256
M[('g', 'g')] = 151 / 256
M[('g', 'b')] = 28 / 256
M[('b', 'r')] = 77 / 256
M[('b', 'g')] = 151 / 256
M[('b', 'b')] = 28 / 256
return M
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
from matutil import identity,mat2coldict,mat2rowdict
from triangular import triangular_solve_n
identity_matrix = identity(A.D[0],one)
identity_dict = mat2coldict(identity_matrix)
rowsdict = mat2rowdict(A)
rowslist = [ rowsdict[i] for i in rowsdict.keys() ]
return coldict2mat([ triangular_solve_n(rowslist,identity_dict[i]) for i in identity_dict.keys() ])
def find_matrix_inverse(A):
'''
Input:
- A: an invertible Mat over GF(2)
Output:
- A Mat that is the inverse of A
Examples:
>>> M1 = Mat(({0,1,2}, {0,1,2}), {(0, 1): one, (1, 0): one, (2, 2): one})
>>> find_matrix_inverse(M1) == Mat(M1.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
>>> M2 = Mat(({0,1,2,3},{0,1,2,3}),{(0,1):one,(1,0):one,(2,2):one})
>>> find_matrix_inverse(M2) == Mat(M2.D, {(0, 1): one, (1, 0): one, (2, 2): one})
True
'''
cols = range(len(mat2coldict(A)))
iden = mat2coldict(identity(A.D[0], one))
B = list()
for n in cols:
ones = iden[n]
B.append(solve(A, ones))
return coldict2mat(B)
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1,
2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3,
3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2):
-0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1,
(2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
L = []
label_list = []
I = identity(A.D[0], 1)
R = mat2rowdict(A)
w = mat2rowdict(I)
label_list.extend(range(len(R)))
for k, v in w.items():
s = triangular_solve(R, label_list, v)
L.append(s)
return coldict2mat(L)
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
from matutil import identity, mat2coldict, mat2rowdict
from triangular import triangular_solve_n
identity_matrix = identity(A.D[0], one)
identity_dict = mat2coldict(identity_matrix)
rowsdict = mat2rowdict(A)
rowslist = [rowsdict[i] for i in rowsdict.keys()]
return coldict2mat([
triangular_solve_n(rowslist, identity_dict[i])
for i in identity_dict.keys()
])
def find_triangular_matrix_inverse(A):
'''
input: An upper triangular Mat, A, with nonzero diagonal elements
output: Inverse of A
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1,
2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3,
3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2):
-0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1,
(2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
L=[]
label_list = []
I = identity(A.D[0],1)
R = mat2rowdict(A)
w = mat2rowdict(I)
label_list.extend(range(len(R)))
for k,v in w.items():
s = triangular_solve(R, label_list, v)
L.append(s)
return coldict2mat(L)
def find_triangular_matrix_inverse(A):
'''
Supporting GF2 is not required.
Input:
- A: an upper triangular Mat with nonzero diagonal elements
Output:
- Mat that is the inverse of A
Example:
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
rows = range(len(mat2rowdict(A)))
b = mat2coldict(identity(A.D[0], 1))
rowlist = mat2rowdict(A)
ret = list()
for n in rows:
ones = b[n]
ret.append(triangular_solve(list(rowlist.values()), list(A.D[1]), ones))
return coldict2mat(ret)
def find_matrix_inverse(A):
return coldict2mat(
[solve(A, b) for (a, b) in mat2coldict(identity(A.D[0], one)).items()])
def find_eigenvector(A, eigenvalue):
A_eigenvalue = A - eigenvalue * identity(A.D)
return find_null_basis(A_eigenvalue)
