SEP/LinearAlgebra - Sagemath Wiki

This may be more of a work-in-progress than a formal SEP, but I\'m not sure where else to put it.

Basically, we now have inconsistency of what functions are defined for what matrix types. This makes things confusing to students, for example; when they are doing a problem, often a matrix will be coerced into a new type, which then has a different set of functions.

We should also carefully look at the eig* functions in each datatype and make sure that the interface is consistent and that the eigenvalues/vectors/spaces/matrices are computed in the appropriate way.

These methods are fine the way they are now:

method	Integer Ring	Rational Field	Real Field with 53 bits of precision	Complex Field with 53 bits of precision	Real Double Field	Complex Double Field	Symbolic Ring
BKZ	TRUE
LLL	TRUE
LLL_gram	TRUE
add_multiple_of_column	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
add_multiple_of_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
apply_map	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
apply_morphism	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
arguments							TRUE
change_ring	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
column	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
column_module	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
commutator	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
copy	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
db	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Is this used??
dense_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
dense_matrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
dense_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
det	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
determinant	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
dict	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
dump	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
dumps	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

These methods still need to be combed through to see if something needs to be done.

method	Integer Ring	Rational Field	Real Field with 53 bits of precision	Complex Field with 53 bits of precision	Real Double Field	Complex Double Field	Symbolic Ring
expand							TRUE
factor							TRUE
fcp	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
find	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
frobenius	TRUE
gcd	TRUE
get_subdivisions	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
gram_schmidt	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
hadamard_bound	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
height	TRUE	TRUE
hermite_form	TRUE
hessenberg_form	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
hessenbergize	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
image	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
index_in_saturation	TRUE
insert_row	TRUE
integer_kernel	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
inverse	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
invert		TRUE
is_LLL_reduced	TRUE
is_dense	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_immutable	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_invertible	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_mutable	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_nilpotent	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_one	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_scalar	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_simplified							TRUE
is_sparse	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_square	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_symmetric	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_unit	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
is_zero	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
iterates	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
jordan_form	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
kernel	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
kernel_matrix	TRUE
kernel_on	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
left_eigenmatrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
left_eigenvectors	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
left_kernel	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
left_nullity	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
lift	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
linear_combination_of_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
linear_combination_of_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
list	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
log_determinant					TRUE	TRUE
matrix_from_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
matrix_from_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
matrix_from_rows_and_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
matrix_over_field	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
matrix_space	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
matrix_window	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
maxspin	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
minimal_polynomial	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
minors	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
minpoly	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
mod	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
multiplicative_order	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
n	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
ncols	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
new_matrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nonpivots	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nonzero_positions	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nonzero_positions_in_column	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nonzero_positions_in_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
norm	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nrows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
nullity	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
number_of_arguments							TRUE
numerical_approx	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
numpy	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
order	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
parent	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
permanent	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
permanental_minor	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
pivot_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
pivots	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
plot	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
prod_of_row_sums	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
randomize	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rank	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rational_reconstruction	TRUE
rename	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rescale_col	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rescale_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
reset_name	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
restrict	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
restrict_codomain	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
restrict_domain	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
right_eigenmatrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
right_eigenspaces	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
right_eigenvectors	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
right_kernel	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
right_nullity	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rook_vector	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
row_module	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
row_space	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
saturation	TRUE
save	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_block	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_col_to_multiple_of_col	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_column	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_immutable	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
set_row_to_multiple_of_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
simplify							TRUE
simplify_rational							TRUE
simplify_trig							TRUE
smith_form	TRUE
solve_left	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
solve_left_LU					TRUE	TRUE
solve_right	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
sparse_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
sparse_matrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
sparse_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
stack	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
str	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
subdivide	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
subdivision	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
subdivision_entry	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
subdivisions	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
submatrix	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
subs	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
substitute	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
swap_columns	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
swap_rows	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
symplectic_form	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
tensor_product	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
trace	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
transpose	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
variables							TRUE
version	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
visualize_structure	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
wiedemann	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_added_multiple_of_column	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_added_multiple_of_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_col_set_to_multiple_of_col	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_rescaled_col	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_rescaled_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
with_row_set_to_multiple_of_row	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

These functions are on the todo list.

method	Integer Ring	Rational Field	Real Field with 53 bits of precision	Complex Field with 53 bits of precision	Real Double Field	Complex Double Field	Symbolic Ring	TODO
LU					TRUE	TRUE
LU_valid					TRUE	TRUE
QR					TRUE	TRUE
SVD					TRUE	TRUE
abs	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Make it clear that this returns the determinant
act_on_polynomial	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Does not deal with symbolic polynomials
additive_order	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Is this implemented for any matrix
adjoint	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Rename to adjugate, deprecate this function. Later, define this function to be the conjugate transpose
antitranspose	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Define antitranspose in the docs
augment	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Make it clear that other will be coerced to a matrix over self.base_ring(). Maybe this should be changed so that the returned matrix is of a type that both matrices can be coerced to?
base_extend	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	No documentation
base_ring	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	No documentation
block_sum	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Same comment as augment above. In fact, fixing augment may fix this.
category	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	No documentation
characteristic_polynomial	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Docs point to charpoly, but in reality, the long name should have all the docs and charpoly should just be an alias
charpoly	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	No documentation explaining the characteristic polynomial.
cholesky					TRUE	TRUE
column_space	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Make this just an alias for column_module. In fact, we might deprecate this, since currently it says it returns a vector space for an integer matrix, for example, but does not (returns a free module).
conjugate	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Does not work for integer matrices (see #4494). Maybe we need to coerce to complex numbers first?
decomposition	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Doesn't work for SR matrices
decomposition_of_subspace	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Doesn't work for SR matrices?
denominator	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Doesn't work for SR matrices
density	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	typo in docstring "ration" -> "ratio"
derivative	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Doesn't work for integer matrices (need to define a derivative function for numeric constants?)
echelon_form	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Implement #3211
echelonize	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Implement #3211
eigenmatrix_left	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	For all the eigenfunctions below, make sure that the return values are consistent across rings.
eigenmatrix_right	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
eigenspaces	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
eigenspaces_left	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
eigenspaces_right	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
eigenvalues	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
eigenvectors_left	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Implement #4834
eigenvectors_right	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	Implement #4834
elementary_divisors	TRUE							Doesn't work over SR matrices; apparently this function was expanded in 3.2.2 to work over more matrices.
exp	#4733	#4733	#4733	#4733	#4733	#4733	TRUE

I got this from this code:

import inspect
rings = [ZZ,QQ,RR,CC,RDF,CDF,SR]

ring_methods = {}

for r in rings:
    ring_methods[r] = set([method for method,_ in inspect.getmembers(matrix(r)) if not method.startswith('_')])

# Get a comprehensive list of names.
full_list = set([])

for r in rings:
    full_list.update(ring_methods[r])

full_list = sorted(list(full_list))

s = '|| method || '
s += ' || '.join([repr(r) for r in rings])+' ||'
print s

for method in full_list:
    s = "|| "+ method+" || "
    s += ' || '.join(['TRUE' if method in ring_methods[r] else '' for r in rings])
    s += ' ||'
    print s