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 | 1 | initial version |
This issue is not really about matrices but about how Sage not being able to figure out that sqrt() here are real numbers that are invariant with respect to conjugate(). It may be possible to help Sage understand this with a set of assumptions, but I do not know how to do so (I've tried assume(norm(w_2) < 1) and alike, but it did not help).
Below is an alternative solution using a function that manually looks for patterns conjugate(sqrt(...)) and replaces them with just sqrt(...)
def simplify_conjugate_of_sqrt(f):
x = SR.wild(0)
return f.subs({conjugate(sqrt(x)):sqrt(x)}).full_simplify()
It amounts only to replacing T.simplify_full() in your code to T.apply_map(simplify_conjugate_of_sqrt) to a much simpler matrix:
[ -(w_2*conjugate(w_2) - 1)*w_3*w_4*conjugate(w_3)*conjugate(w_4) -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_4*conjugate(w_3)*conjugate(w_4) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0]
[ -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_3*w_4*conjugate(w_4) -(w_3*conjugate(w_3) - 1)*w_4*conjugate(w_4) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0]
[ sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*w_4 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_4 -w_4*conjugate(w_4) + 1 0]
[ 0 0 0 1]
Basically it matches the OP's expected matrix, with just each norm(z) replaced with z * conjugate(z), but this is how Sage handles the norm(z) function.
| | 2 | No.2 Revision |
This issue is not really about matrices but about how Sage not being able to figure out that sqrt() here are real numbers that are invariant with respect to conjugate(). It may be possible to help Sage understand this with a set of assumptions, but I do not know how to do so (I've tried assume(norm(w_2) < 1) and alike, but it did not help).
Below is an alternative solution using a function that manually looks for patterns conjugate(sqrt(...)) and replaces them with just sqrt(...)
def simplify_conjugate_of_sqrt(f):
x = SR.wild(0)
return f.subs({conjugate(sqrt(x)):sqrt(x)}).full_simplify()
It amounts only to replacing T.simplify_full() in your OP's code to T.apply_map(simplify_conjugate_of_sqrt) to get a much simpler matrix:
[ -(w_2*conjugate(w_2) - 1)*w_3*w_4*conjugate(w_3)*conjugate(w_4) -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_4*conjugate(w_3)*conjugate(w_4) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0]
[ -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_3*w_4*conjugate(w_4) -(w_3*conjugate(w_3) - 1)*w_4*conjugate(w_4) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0]
[ sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*w_4 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_4 -w_4*conjugate(w_4) + 1 0]
[ 0 0 0 1]
Basically it matches the OP's expected matrix, with just each norm(z) replaced with z * conjugate(z), but this is how Sage handles the norm(z) function.
| | 3 | No.3 Revision |
This issue is not really about matrices but about how Sage not being able to figure out that sqrt() here are real numbers that are invariant with respect to conjugate(). It may be possible to help Sage understand this with a set of assumptions, but I do not know how to do so (I've tried assume(norm(w_2) < 1) and alike, but it did not help).
Below is an alternative solution using a function that manually looks for patterns conjugate(sqrt(...)) and replaces them with just sqrt(...)
def simplify_conjugate_of_sqrt(f):
x = SR.wild(0)
return f.subs({conjugate(sqrt(x)):sqrt(x)}).full_simplify()
It amounts only to replacing T.simplify_full() in OP's code to with T.apply_map(simplify_conjugate_of_sqrt) to get a much simpler matrix:
[ -(w_2*conjugate(w_2) - 1)*w_3*w_4*conjugate(w_3)*conjugate(w_4) -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_4*conjugate(w_3)*conjugate(w_4) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0]
[ -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_3*w_4*conjugate(w_4) -(w_3*conjugate(w_3) - 1)*w_4*conjugate(w_4) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0]
[ sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*w_4 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_4 -w_4*conjugate(w_4) + 1 0]
[ 0 0 0 1]
Basically it matches the OP's expected matrix, with just each norm(z) replaced with z * conjugate(z), but this is how Sage handles the norm(z) function.
| | 4 | No.4 Revision |
This issue is not really about matrices but about how Sage not being able to figure out that sqrt() here are real numbers that are invariant with respect to conjugate(). It may be possible to help Sage understand this with a set of assumptions, but I do not know how to do so (I've tried assume(norm(w_2) < 1) and alike, but it did not help).
Below is I present an alternative solution solution, using a function that manually looks for patterns conjugate(sqrt(...)) and in a given expression. replaces them with just sqrt(...), and simplifies the result:
def simplify_conjugate_of_sqrt(f):
x = SR.wild(0)
return f.subs({conjugate(sqrt(x)):sqrt(x)}).full_simplify()
It amounts only to replacing T.simplify_full() in OP's code with T.apply_map(simplify_conjugate_of_sqrt) to get a much simpler matrix:
[ -(w_2*conjugate(w_2) - 1)*w_3*w_4*conjugate(w_3)*conjugate(w_4) -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_4*conjugate(w_3)*conjugate(w_4) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0]
[ -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_3*w_4*conjugate(w_4) -(w_3*conjugate(w_3) - 1)*w_4*conjugate(w_4) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0]
[ sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*w_4 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_4 -w_4*conjugate(w_4) + 1 0]
[ 0 0 0 1]
Basically it matches the OP's expected matrix, with just each norm(z) replaced with z * conjugate(z), but this is how Sage handles the norm(z) function.
| | 5 | No.5 Revision |
This issue is not really about matrices but about how Sage not being able to figure out that sqrt() here are real numbers that are invariant with respect to conjugate(). It may be possible to help Sage understand this with a set of assumptions, but I do not know how to do so (I've tried assume(norm(w_2) < 1) and alike, but it did not help).
Below I present an alternative solution, using a the following function that manually looks for patterns conjugate(sqrt(...)) in a given expression. expression, replaces them with just sqrt(...), and simplifies the result:
def simplify_conjugate_of_sqrt(f):
x = SR.wild(0)
return f.subs({conjugate(sqrt(x)):sqrt(x)}).full_simplify()
It amounts only to replacing T.simplify_full() in OP's code with T.apply_map(simplify_conjugate_of_sqrt) to get a much simpler matrix:
[ -(w_2*conjugate(w_2) - 1)*w_3*w_4*conjugate(w_3)*conjugate(w_4) -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_4*conjugate(w_3)*conjugate(w_4) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0]
[ -sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*w_3*w_4*conjugate(w_4) -(w_3*conjugate(w_3) - 1)*w_4*conjugate(w_4) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0]
[ sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*w_4 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_4 -w_4*conjugate(w_4) + 1 0]
[ 0 0 0 1]
Basically it matches the OP's expected matrix, with just each norm(z) replaced with z * conjugate(z), but this is how Sage handles the norm(z) function.
