Module: sage.matrix.matrix_integer_dense
Dense matrices over the integer ring.
Author Log:
sage: a = matrix(ZZ, 3,3, range(9)); a [0 1 2] [3 4 5] [6 7 8] sage: a.det() 0 sage: a[0,0] = 10; a.det() -30 sage: a.charpoly() x^3 - 22*x^2 + 102*x + 30 sage: b = -3*a sage: a == b False sage: b < a True
TESTS:
sage: a = matrix(ZZ,2,range(4), sparse=False) sage: loads(dumps(a)) == a True
Module-level Functions
) |
) |
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Compare various multiplication algorithms.
Input:
sage: from sage.matrix.matrix_integer_dense import tune_multiplication sage: tune_multiplication(2, nmin=10, nmax=60, bitmin=2,bitmax=8) 10 2 0.2 ...
Class: Matrix_integer_dense
On a 32-bit machine, they can have at most
rows or
columns. On a 64-bit machine, matrices can have at most
rows or columns.
sage: a = MatrixSpace(ZZ,3)(2); a [2 0 0] [0 2 0] [0 0 2] sage: a = matrix(ZZ,1,3, [1,2,-3]); a [ 1 2 -3] sage: a = MatrixSpace(ZZ,2,4)(2); a Traceback (most recent call last): ... TypeError: nonzero scalar matrix must be square
Functions: charpoly,
decomposition,
determinant,
echelon_form,
elementary_divisors,
frobenius,
gcd,
height,
hermite_form,
index_in_saturation,
insert_row,
kernel,
kernel_matrix,
LLL,
LLL_gram,
minpoly,
pivots,
prod_of_row_sums,
randomize,
rank,
rational_reconstruction,
saturation,
smith_form,
stack,
symplectic_form
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Input:
NOTE: Linbox charpoly disabled on 64-bit machines, since it hangs in many cases.
sage: A = matrix(ZZ,6, range(36)) sage: f = A.charpoly(); f x^6 - 105*x^5 - 630*x^4 sage: f(A) == 0 True sage: n=20; A = Mat(ZZ,n)(range(n^2)) sage: A.charpoly() x^20 - 3990*x^19 - 266000*x^18 sage: A.minpoly() # optional -- os x only right now x^3 - 3990*x^2 - 266000*x
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Returns the decomposition of the free module on which this matrix A acts from the right (i.e., the action is x goes to x A), along with whether this matrix acts irreducibly on each factor. The factors are guaranteed to be sorted in the same way as the corresponding factors of the characteristic polynomial, and are saturated as ZZ modules.
Input:
sage: t = ModularSymbols(11,sign=1).hecke_matrix(2) sage: w = t.change_ring(ZZ) sage: w.list() [3, -1, 0, -2]
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Return the determinant of this matrix.
Input: algorithm -
ALGORITHM:
The p-adic algorithm works by first finding a random
vector v, then solving A*x = v and taking the denominator
. This gives a divisor of the determinant. Then we
compute
using a multimodular algorithm and the
Hadamard bound, skipping primes that divide
.
TIMINGS: This is perhaps the fastest implementation of determinants in the world. E.g., for a 500x500 random matrix with 32-bit entries on a core2 duo 2.6Ghz running OS X, Sage takes 4.12 seconds, whereas Magma takes 62.87 seconds (both with proof False). With proof=True on the same problem Sage takes 5.73 seconds. For another example, a 200x200 random matrix with 1-digit entries takes 4.18 seconds in pari, 0.18 in Sage with proof True, 0.11 in Sage with proof False, and 0.21 seconds in Magma with proof True and 0.18 in Magma with proof False.
sage: A = matrix(ZZ,8,8,[3..66]) sage: A.determinant() 0
sage: A = random_matrix(ZZ,20,20) sage: D1 = A.determinant() sage: A._clear_cache() sage: D2 = A.determinant(algorithm='ntl') sage: D1 == D2 True
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Return the echelon form of this matrix over the integers also known as the hermit normal form (HNF).
Input: algorithm -
NOTE: The result is not cached.
sage: A = MatrixSpace(ZZ,2)([1,2,3,4]) sage: A.echelon_form() [1 0] [0 2]
sage: A = MatrixSpace(ZZ,5)(range(25)) sage: A.echelon_form() [ 5 0 -5 -10 -15] [ 0 1 2 3 4] [ 0 0 0 0 0] [ 0 0 0 0 0] [ 0 0 0 0 0]
TESTS: Make sure the zero matrices are handled correctly:
sage: m = matrix(ZZ,3,3,[0]*9) sage: m.echelon_form() [0 0 0] [0 0 0] [0 0 0] sage: m = matrix(ZZ,3,1,[0]*3) sage: m.echelon_form() [0] [0] [0] sage: m = matrix(ZZ,1,3,[0]*3) sage: m.echelon_form() [0 0 0]
The ultimate border case!
sage: m = matrix(ZZ,0,0,[]) sage: m.echelon_form() []
NOTE: If 'ntl' is chosen for a non square matrix this function raises a ValueError.
Special cases: 0 or 1 rows:
sage: a = matrix(ZZ, 1,2,[0,-1]) sage: a.hermite_form() [0 1] sage: a.pivots() [1] sage: a = matrix(ZZ, 1,2,[0,0]) sage: a.hermite_form() [0 0] sage: a.pivots() [] sage: a = matrix(ZZ,1,3); a [0 0 0] sage: a.echelon_form(include_zero_rows=False) [] sage: a.echelon_form(include_zero_rows=True) [0 0 0]
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Return the elementary divisors of self, in order.
IMPLEMENTATION: Uses linbox, except sometimes linbox doesn't work (errors about pre-conditioning), in which case PARI is used.
WARNING: This is MUCH faster than the smith_form function.
The elementary divisors are the invariants of the finite abelian group that is the cokernel of this matrix. They are ordered in reverse by divisibility.
Input:
sage: matrix(3, range(9)).elementary_divisors() [1, 3, 0] sage: matrix(3, range(9)).elementary_divisors(algorithm='pari') [1, 3, 0] sage: C = MatrixSpace(ZZ,4)([3,4,5,6,7,3,8,10,14,5,6,7,2,2,10,9]) sage: C.elementary_divisors() [1, 1, 1, 687]
sage: M = matrix(ZZ, 3, [1,5,7, 3,6,9, 0,1,2]) sage: M.elementary_divisors() [1, 1, 6]
This returns a copy, which is safe to change:
sage: edivs = M.elementary_divisors() sage: edivs.pop() 6 sage: M.elementary_divisors() [1, 1, 6]
SEE ALSO: smith_form
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Return the Frobenius form (rational canonical form) of this matrix.
Input: flag -an integer:
Input:
ALGORITHM: uses pari's matfrobenius()
sage: A = MatrixSpace(ZZ, 3)(range(9)) sage: A.frobenius(0) [ 0 0 0] [ 1 0 18] [ 0 1 12] sage: A.frobenius(1) [x^3 - 12*x^2 - 18*x] sage: A.frobenius(1, var='y') [y^3 - 12*y^2 - 18*y] sage: A.frobenius(2) ([ 0 0 0] [ 1 0 18] [ 0 1 12], [ -1 2 -1] [ 0 23/15 -14/15] [ 0 -2/15 1/15]) sage: a=matrix([]) sage: a.frobenius(2) ([], []) sage: a.frobenius(0) [] sage: a.frobenius(1) [] sage: B = random_matrix(ZZ,2,3) sage: B.frobenius() Traceback (most recent call last): ... ArithmeticError: frobenius matrix of non-square matrix not defined.
Author: 2006-04-02: Martin Albrecht
TODO: - move this to work for more general matrices than just over Z. This will require fixing how PARI polynomials are coerced to SAGE polynomials.
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Return the gcd of all entries of self; very fast.
sage: a = matrix(ZZ,2, [6,15,-6,150]) sage: a.gcd() 3
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Return the height of this matrix, i.e., the max absolute value of the entries of the matrix.
Output: A nonnegative integer.
sage: a = Mat(ZZ,3)(range(9)) sage: a.height() 8 sage: a = Mat(ZZ,2,3)([-17,3,-389,15,-1,0]); a [ -17 3 -389] [ 15 -1 0] sage: a.height() 389
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Return the Hermite normal form of self.
This is a synonym for self.echelon_form(...)
. See the documentation
for self.echelon_form
for more details.
sage: A = matrix(ZZ, 3, 5, [-1, -1, -2, 2, -2, -4, -19, -17, 1, 2, -3, 1, 1, -4, 1]) sage: E, U = A.hermite_form(transformation=True) sage: E [ 1 0 52 -133 109] [ 0 1 19 -47 38] [ 0 0 69 -178 145] sage: U [-46 3 11] [-16 1 4] [-61 4 15] sage: U*A [ 1 0 52 -133 109] [ 0 1 19 -47 38] [ 0 0 69 -178 145] sage: A.hermite_form() [ 1 0 52 -133 109] [ 0 1 19 -47 38] [ 0 0 69 -178 145]
TESTS: This example illustrated trac 2398.
sage: a = matrix([(0, 0, 3), (0, -2, 2), (0, 1, 2), (0, -2, 5)]) sage: a.hermite_form() [0 1 2] [0 0 3] [0 0 0] [0 0 0]
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Return the index of self in its saturation.
Input:
ALGORITHM: Use Hermite normal form twice to find an invertible matrix whose inverse transforms a matrix with the same row span as self to its saturation, then compute the determinant of that matrix.
sage: A = matrix(ZZ, 2,3, [1..6]); A [1 2 3] [4 5 6] sage: A.index_in_saturation() 3 sage: A.saturation() [1 2 3] [1 1 1]
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Create a new matrix from self with.
Input:
sage: X = matrix(ZZ,3,range(9)); X [0 1 2] [3 4 5] [6 7 8] sage: X.insert_row(1, [1,5,-10]) [ 0 1 2] [ 1 5 -10] [ 3 4 5] [ 6 7 8] sage: X.insert_row(0, [1,5,-10]) [ 1 5 -10] [ 0 1 2] [ 3 4 5] [ 6 7 8] sage: X.insert_row(3, [1,5,-10]) [ 0 1 2] [ 3 4 5] [ 6 7 8] [ 1 5 -10]
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Return the left kernel of this matrix, as a module over the integers. This is the saturated ZZ-module spanned by all the row vectors v such that v*self = 0.
Input:
By convention if self has 0 rows, the kernel is of dimension 0, whereas the kernel is the whole domain if self has 0 columns.
sage: M = MatrixSpace(ZZ,4,2)(range(8)) sage: M.kernel() Free module of degree 4 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 -3 2] [ 0 1 -2 1]
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The options are exactly like self.kernel(...), but returns a matrix A whose rows form a basis for the left kernel, i.e., so that A*self = 0.
This is mainly useful to avoid all overhead associated with creating a free module.
sage: A = matrix(ZZ, 3, 3, [1..9]) sage: A.kernel_matrix() [-1 2 -1]
Note that the basis matrix returned above is not in Hermite form.
sage: A.kernel() Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 1 -2 1]
We compute another kernel:
sage: A = matrix(ZZ, 4, 2, [2, -1, 1, 1, -18, -1, -1, -5]) sage: K = A.kernel_matrix(); K [-17 -20 -3 0] [ 7 3 1 -1]
K is a basis for the left kernel:
sage: K*A [0 0] [0 0]
We illustrate the LLL flag:
sage: L = A.kernel_matrix(LLL=True); L [ 7 3 1 -1] [ 4 -11 0 -3] sage: K.hermite_form() [ 1 64 3 12] [ 0 89 4 17] sage: L.hermite_form() [ 1 64 3 12] [ 0 89 4 17]
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Returns LLL reduced or approximated LLL reduced lattice R for this matrix interpreted as a lattice.
A lattice
is
-LLL-reduced
if the two following conditions hold:
(a) For any
, we have
,
(b) For any
, we have
,
where
and
is
the
-th vector of the Gram-Schmidt orthogonalisation of
.
The default reduction parameters are
and
. The parameters
and
must satisfy:
and
. Polynomial time complexity is only guaranteed
for
.
The lattice is returned as a matrix. Also the rank (and the determinant) of self are cached if those are computed during the reduction. Note that in general this only happens when self.rank() == self.ncols() and the exact algorithm is used.
Input:
Also, if the verbose level is >= 2, some more verbose output is printed during the calculation if NTL is used.
AVAILABLE ALGORITHMS:
sage: A = Matrix(ZZ,3,3,range(1,10)) sage: A.LLL() [ 0 0 0] [ 2 1 0] [-1 1 3]
We compute the extended GCD of a list of integers using LLL, this example is from the Magma handbook:
sage: Q = [ 67015143, 248934363018, 109210, 25590011055, 74631449, \ 10230248, 709487, 68965012139, 972065, 864972271 ] sage: n = len(Q) sage: S = 100 sage: X = Matrix(ZZ, n, n + 1) sage: for i in xrange(n): ... X[i,i + 1] = 1 sage: for i in xrange(n): ... X[i,0] = S*Q[i] sage: L = X.LLL() sage: M = L.row(n-1).list()[1:] sage: M [-3, -1, 13, -1, -4, 2, 3, 4, 5, -1] sage: add([Q[i]*M[i] for i in range(n)]) -1
ALGORITHM: Uses the NTL library by Victor Shoup or fpLLL library by Damien Stehle depending on the chosen algorithm.
REFERENCES: ntl.mat_ZZ
or sage.libs.fplll.fplll
for details on the used algorithms.
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LLL reduction of the lattice whose gram matrix is self.
Input:
U.transpose() * M * U
is LLL-reduced
ALGORITHM: Use PARI
sage: M = Matrix(ZZ, 2, 2, [5,3,3,2]) ; M [5 3] [3 2] sage: U = M.LLL_gram(); U [-1 1] [ 1 -2] sage: U.transpose() * M * U [1 0] [0 1]
Semidefinite and indefinite forms raise a ValueError:
sage: Matrix(ZZ,2,2,[2,6,6,3]).LLL_gram() Traceback (most recent call last): ... ValueError: not a definite matrix sage: Matrix(ZZ,2,2,[1,0,0,-1]).LLL_gram() Traceback (most recent call last): ... ValueError: not a definite matrix
BUGS: should work for semidefinite forms (PARI is ok)
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Input:
NOTE: Linbox charpoly disabled on 64-bit machines, since it hangs in many cases.
sage: A = matrix(ZZ,6, range(36)) sage: A.minpoly() # optional -- os x only right now x^3 - 105*x^2 - 630*x sage: n=6; A = Mat(ZZ,n)([k^2 for k in range(n^2)]) sage: A.minpoly() # optional -- os x only right now x^4 - 2695*x^3 - 257964*x^2 + 1693440*x
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Return the pivot column positions of this matrix as a list of Python integers.
This returns a list, of the position of the first nonzero entry in each row of the echelon form.
Output:
sage: n = 3; A = matrix(ZZ,n,range(n^2)); A [0 1 2] [3 4 5] [6 7 8] sage: A.pivots() [0, 1] sage: A.echelon_form() [ 3 0 -3] [ 0 1 2] [ 0 0 0]
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Return the product of the sums of the entries in the submatrix of self with given columns.
Input:
sage: a = matrix(ZZ,2,3,[1..6]); a [1 2 3] [4 5 6] sage: a.prod_of_row_sums([0,2]) 40 sage: (1+3)*(4+6) 40 sage: a.prod_of_row_sums(set([0,2])) 40
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Randomize density proportion of the entries of this matrix, leaving the rest unchanged.
The parameters are the same as the integer ring's random_element function.
If x and y are given, randomized entries of this matrix to be between x and y and have density 1.
Input:
sage: A = matrix(ZZ, 2,3, [1..6]); A [1 2 3] [4 5 6] sage: A.randomize() sage: A [-8 2 0] [ 0 1 -1] sage: A.randomize(x=-30,y=30) sage: A [ 5 -19 24] [ 24 23 -9]
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Return the rank of this matrix.
Output:
NOTE: The rank is cached.
ALGORITHM: First check if the matrix has maxim posible rank by working modulo one random prime. If not call Linbox's rank function.
sage: a = matrix(ZZ,2,3,[1..6]); a [1 2 3] [4 5 6] sage: a.rank() 2 sage: a = matrix(ZZ,3,3,[1..9]); a [1 2 3] [4 5 6] [7 8 9] sage: a.rank() 2
Here's a bigger example - the rank is of course still 2:
sage: a = matrix(ZZ,100,[1..100^2]); a.rank() 2
) |
Use rational reconstruction to lift self to a matrix over the rational numbers (if possible), where we view self as a matrix modulo N.
Input:
We create a random 4x4 matrix over ZZ.
sage: A = matrix(ZZ, 4, [4, -4, 7, 1, -1, 1, -1, -12, -1, -1, 1, -1, -3, 1, 5, -1])
There isn't a unique rational reconstruction of it:
sage: A.rational_reconstruction(11) Traceback (most recent call last): ... ValueError: Rational reconstruction of 4 (mod 11) does not exist.
We throw in a denominator and reduce the matrix modulo 389 - it does rationally reconstruct.
sage: B = (A/3 % 389).change_ring(ZZ) sage: B.rational_reconstruction(389) == A/3 True
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Return a saturation matrix of self, which is a matrix whose rows span the saturation of the row span of self. This is not unique.
The saturation of a
module
embedded in
is
the a module
that contains
with finite index such that
is torsion free. This function takes the row span
of self, and finds another matrix of full rank with row
span the saturation of
.
Input:
NOTE: The result is not cached.
ALGORITHM:
1. Replace input by a matrix of full rank got from a
subset of the rows.
2. Divide out any common factors from rows.
3. Check max_dets random dets of submatrices to see if their
gcd (with p) is 1 - if so matrix is saturated and we're done.
4. Finally, use that if A is a matrix of full rank, then
is a saturation of A.
sage: A = matrix(ZZ, 3, 5, [-51, -1509, -71, -109, -593, -19, -341, 4, 86, 98, 0, -246, -11, 65, 217]) sage: A.echelon_form() [ 1 5 2262 20364 56576] [ 0 6 35653 320873 891313] [ 0 0 42993 386937 1074825] sage: S = A.saturation(); S [ -51 -1509 -71 -109 -593] [ -19 -341 4 86 98] [ 35 994 43 51 347]
Notice that the saturation spans a different module than A.
sage: S.echelon_form() [ 1 2 0 8 32] [ 0 3 0 -2 -6] [ 0 0 1 9 25] sage: V = A.row_space(); W = S.row_space() sage: V.is_submodule(W) True sage: V.index_in(W) 85986 sage: V.index_in_saturation() 85986
We illustrate each option:
sage: S = A.saturation(p=2) sage: S = A.saturation(proof=False) sage: S = A.saturation(max_dets=2)
) |
Returns matrices S, U, and V such that S = U*self*V, and S is in Smith normal form. Thus S is diagonal with diagonal entries the ordered elementary divisors of S.
WARNING: The elementary_divisors function, which returns the diagonal entries of S, is VASTLY faster than this function.
The elementary divisors are the invariants of the finite abelian group that is the cokernel of this matrix. They are ordered in reverse by divisibility.
sage: A = MatrixSpace(IntegerRing(), 3)(range(9)) sage: D, U, V = A.smith_form() sage: D [0 0 0] [0 3 0] [0 0 1] sage: U [-1 2 -1] [ 0 -1 1] [ 0 1 0] sage: V [ 1 4 -1] [-2 -3 1] [ 1 0 0] sage: U*A*V [0 0 0] [0 3 0] [0 0 1]
It also makes sense for nonsquare matrices:
sage: A = Matrix(ZZ,3,2,range(6)) sage: D, U, V = A.smith_form() sage: D [0 0] [2 0] [0 1] sage: U [-1 2 -1] [ 0 -1 1] [ 0 1 0] sage: V [ 3 -1] [-2 1] sage: U * A * V [0 0] [2 0] [0 1]
SEE ALSO: elementary_divisors
) |
Return the matrix self on top of other: [ self ] [ other ]
sage: M = Matrix(ZZ, 2, 3, range(6)) sage: N = Matrix(ZZ, 1, 3, [10,11,12]) sage: M.stack(N) [ 0 1 2] [ 3 4 5] [10 11 12]
) |
Find a symplectic basis for self if self is an anti-symmetric, alternating matrix.
Returns a pair (F, C) such that the rows of C form a symplectic basis for self and F = C * self * C.transpose().
Raises a ValueError if self is not anti-symmetric, or self is not alternating.
Anti-symmetric means that
. Alternating means that the
diagonal of
is identically zero.
A symplectic basis is a basis of the form
such that
*
= 0 for all vectors
;
*
for all
;
*
for all
;
*
for all
, and
for all
;
*
for all
not equal
.
The ordering for the factors
and for the
placement of zeroes was chosen to agree with the output of
smith_form
.
See the example for a pictorial description of such a basis.
sage: E = matrix(ZZ, 5, 5, [0, 14, 0, -8, -2, -14, 0, -3, -11, 4, 0, 3, 0, 0, 0, 8, 11, 0, 0, 8, 2, -4, 0, -8, 0]); E [ 0 14 0 -8 -2] [-14 0 -3 -11 4] [ 0 3 0 0 0] [ 8 11 0 0 8] [ 2 -4 0 -8 0] sage: F, C = E.symplectic_form() sage: F [ 0 0 0 0 0] [ 0 0 0 2 0] [ 0 0 0 0 1] [ 0 -2 0 0 0] [ 0 0 -1 0 0] sage: F == C * E * C.transpose() True sage: E.smith_form()[0] [0 0 0 0 0] [0 2 0 0 0] [0 0 2 0 0] [0 0 0 1 0] [0 0 0 0 1]
Special Functions: __copy__,
__eq__,
__ge__,
__gt__,
__init__,
__le__,
__lt__,
__ne__,
_add_row_and_maintain_echelon_form,
_adjoint,
_change_ring,
_charpoly_linbox,
_delete_zero_columns,
_det_linbox,
_det_ntl,
_echelon_in_place_classical,
_echelon_strassen,
_elementary_divisors_linbox,
_factor_out_common_factors_from_each_row,
_hnf_mod,
_insert_zero_columns,
_invert_iml,
_kernel_gens_using_pari,
_kernel_matrix_using_padic_algorithm,
_linbox_dense,
_linbox_modn,
_linbox_modn_det,
_linbox_sparse,
_minpoly_linbox,
_mod_int,
_multiply_classical,
_multiply_linbox,
_multiply_multi_modular,
_ntl_,
_pickle,
_poly_linbox,
_rank_linbox,
_rank_modp,
_rational_echelon_via_solve,
_rational_kernel_iml,
_reduce,
_solve_iml,
_solve_right_nonsingular_square,
_unpickle
) |
Returns a new copy of this matrix.
sage: a = matrix(ZZ,1,3, [1,2,-3]); a [ 1 2 -3] sage: b = a.__copy__(); b [ 1 2 -3] sage: b is a False sage: b == a True
) |
Assuming self is a full rank n x m matrix in reduced row Echelon form over ZZ and row is a vector of degree m, this function creates a new matrix that is the echelon form of self with row appended to the bottom.
WARNING: It is assumed that self is in echelon form.
Input:
ALGORITHM: For each pivot column of self, we use the extended Euclidean algorithm to clear the column. The result is a new matrix B whose row span is the same as self.stack(row), and whose last row is 0 if and only if row is in the QQ-span of the rows of self. If row is not in the QQ-span of the rows of self, then row is nonzero and suitable to be inserted into the top n rows of A to form a new matrix that is in reduced row echelon form. We then clear that corresponding new pivot column.
sage: a = matrix(ZZ, 3, [1, 0, 110, 0, 3, 112, 0, 0, 221]); a [ 1 0 110] [ 0 3 112] [ 0 0 221] sage: a._add_row_and_maintain_echelon_form(vector(ZZ,[1,2,3]),[0,1,2]) ([1 0 0] [0 1 0] [0 0 1], [0, 1, 2]) sage: a._add_row_and_maintain_echelon_form(vector(ZZ,[0,0,0]),[0,1,2]) ([ 1 0 110] [ 0 3 112] [ 0 0 221], [0, 1, 2]) sage: a = matrix(ZZ, 2, [1, 0, 110, 0, 3, 112]) sage: a._add_row_and_maintain_echelon_form(vector(ZZ,[1,2,3]),[0,1]) ([ 1 0 110] [ 0 1 219] [ 0 0 545], [0, 1, 2])
) |
Return the adjoint of this matrix.
Assumes self is a square matrix (checked in adjoint).
) |
Return the matrix obtained by coercing the entries of this matrix into the given ring.
sage: a = matrix(ZZ,2,[1,-7,3,5]) sage: a._change_ring(RDF) [ 1.0 -7.0] [ 3.0 5.0]
) |
) |
Return matrix obtained from self by deleting all zero columns along with the positions of those columns.
Output: matrix list of integers
sage: a = matrix(ZZ, 2,3, [1,0,3,-1,0,5]); a [ 1 0 3] [-1 0 5] sage: a._delete_zero_columns() ([ 1 3] [-1 5], [1])
) |
Compute the determinant of this matrix using Linbox.
) |
Compute the determinant of this matrix using NTL.
) |
) |
) |
) |
Very very quickly modifies self so that the gcd of the entries in each row is 1 by dividing each row by the common gcd.
sage: a = matrix(ZZ, 3, [-9, 3, -3, -36, 18, -5, -40, -5, -5, -20, -45, 15, 30, -15, 180]) sage: a [ -9 3 -3 -36 18] [ -5 -40 -5 -5 -20] [-45 15 30 -15 180] sage: a._factor_out_common_factors_from_each_row() sage: a [ -3 1 -1 -12 6] [ -1 -8 -1 -1 -4] [ -3 1 2 -1 12]
) |
Input:
) |
Return matrix obtained by self by inserting zero columns so that the columns with positions specified in cols are all 0.
Input:
sage: a = matrix(ZZ, 2,3, [1,0,3,-1,0,5]); a [ 1 0 3] [-1 0 5] sage: b, cols = a._delete_zero_columns() sage: b [ 1 3] [-1 5] sage: cols [1] sage: b._insert_zero_columns(cols) [ 1 0 3] [-1 0 5]
) |
Invert this matrix using IML. The output matrix is an integer matrix and a denominator.
Input:
ALGORITHM: Uses IML's p-adic nullspace function.
sage: a = matrix(ZZ,3,[1,2,5, 3,7,8, 2,2,1]) sage: b, d = a._invert_iml(); b,d ([ 9 -8 19] [-13 9 -7] [ 8 -2 -1], 23) sage: a*b [23 0 0] [ 0 23 0] [ 0 0 23]
Author: William Stein
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Compute an LLL reduced list of independent generators that span the kernel of self.
ALGORITHM: Call pari's matkerint function.
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Compute a list of independent generators that span the right kernel of self.
ALGORITHM: Use IML to compute the kernel over QQ, clear denominators, then saturate.
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Return modn linbox object associated to this integer matrix.
sage: a = matrix(ZZ, 3, [1,2,5,-7,8,10,192,5,18]) sage: b = a._linbox_modn(19); b <sage.libs.linbox.linbox.Linbox_modn_dense object at ...> sage: b.charpoly() [2L, 10L, 11L, 1L]
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Input:
sage: a = matrix(ZZ, 3, [1,2,5,-7,8,10,192,5,18]) sage: a.det() -3669 sage: a._linbox_modn_det(5077) 1408 sage: a._linbox_modn_det(3) 0 sage: a._linbox_modn_det(2) 1 sage: a.det()%5077 1408 sage: a.det()%2 1 sage: a.det()%3 0
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) |
) |
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sage: n = 3 sage: a = MatrixSpace(ZZ,n,n)(range(n^2)) sage: b = MatrixSpace(ZZ,n,n)(range(1, n^2 + 1)) sage: a._multiply_classical(b) [ 18 21 24] [ 54 66 78] [ 90 111 132]
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Multiply matrices over ZZ using linbox.
WARNING: This is very slow right now, i.e., linbox is very slow.
sage: A = matrix(ZZ,2,3,range(6)) sage: A*A.transpose() [ 5 14] [14 50] sage: A._multiply_linbox(A.transpose()) [ 5 14] [14 50]
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ntl.mat_ZZ representation of self.
sage: a = MatrixSpace(ZZ,200).random_element(x=-2, y=2) # -2 to 2 sage: A = a._ntl_()
Note: NTL only knows dense matrices, so if you provide a sparse matrix NTL will allocate memory for every zero entry.
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sage: S = ModularSymbols(250,4,sign=1).cuspidal_submodule().new_subspace().decomposition() # long sage: S == loads(dumps(S)) # long True
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Input:
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Compute the rank of this matrix using Linbox.
) |
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Computes information that gives the reduced row echelon form (over QQ!) of a matrix with integer entries.
Input:
If you put standard basis vectors in order at the pivot columns, and put the matrix (1/d)*X everywhere else, then you get the reduced row echelon form of self, without zero rows at the bottom.
NOTE: IML is the actual underlying
-adic solver that we use.
Author: William Stein
ALGORITHM: I came up with this algorithm from scratch. As far as I know it is new. It's pretty simple, but it is ... (fast?!).
Let A be the input matrix.
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IML: Return the rational kernel of this matrix (acting from the left), considered as a matrix over QQ. I.e., returns a matrix K such that self*K = 0, and the number of columns of K equals the nullity of self.
Author: William Stein
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Let A equal self be a square matrix. Given B return an integer matrix C and an integer d such that self C*A == d*B if right is False or A*C == d*B if right is True.
Output:
sage: A = matrix(ZZ,4,4,[0, 1, -2, -1, -1, 1, 0, 2, 2, 2, 2, -1, 0, 2, 2, 1]) sage: B = matrix(ZZ,3,4, [-1, 1, 1, 0, 2, 0, -2, -1, 0, -2, -2, -2]) sage: C,d = A._solve_iml(B,right=False); C [ 6 -18 -15 27] [ 0 24 24 -36] [ 4 -12 -6 -2]
sage: d 12
sage: C*A == d*B True
sage: A = matrix(ZZ,4,4,[0, 1, -2, -1, -1, 1, 0, 2, 2, 2, 2, -1, 0, 2, 2, 1]) sage: B = matrix(ZZ,4,3, [-1, 1, 1, 0, 2, 0, -2, -1, 0, -2, -2, -2]) sage: C,d = A._solve_iml(B) sage: C [ 12 40 28] [-12 -4 -4] [ -6 -25 -16] [ 12 34 16]
sage: d 12
sage: A*C == d*B True
ALGORITHM: Uses IML.
Author: Martin Albrecht
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If self is a matrix
of full rank, then this function
returns a vector or matrix
such that
. If
is
a vector then
is a vector and if
is a matrix, then
is a matrix. The base ring of
is the integers unless a
denominator is needed in which case the base ring is the
rational numbers.
NOTE: In SAGE one can also write A B
for
A.solve_right(B)
, i.e., SAGE implements the ``the
MATLAB/Octave backslash operator''.
NOTE: This is currently only implemented when A is square.
Input:
sage: a = matrix(ZZ, 2, [0, -1, 1, 0]) sage: v = vector(ZZ, [2, 3]) sage: a \ v (3, -2)
Note that the output vector or matrix is always over
.
sage: parent(a\v) Vector space of dimension 2 over Rational Field
We solve a bigger system where the answer is over the rationals.
sage: a = matrix(ZZ, 3, 3, [1,2,3,4, 5, 6, 8, -2, 3]) sage: v = vector(ZZ, [1,2,3]) sage: w = a \ v; w (2/15, -4/15, 7/15) sage: parent(w) Vector space of dimension 3 over Rational Field sage: a * w (1, 2, 3)
We solve a system where the right hand matrix has multiple columns.
sage: a = matrix(ZZ, 3, 3, [1,2,3,4, 5, 6, 8, -2, 3]) sage: b = matrix(ZZ, 3, 2, [1,5, 2, -3, 3, 0]) sage: w = a \ b; w [ 2/15 -19/5] [-4/15 -27/5] [ 7/15 98/15] sage: a * w [ 1 5] [ 2 -3] [ 3 0]
TESTS: We create a random 100x100 matrix and solve the corresponding system, then verify that the result is correct. (NOTE: This test is very risky without having a seeded random number generator!)
sage: n = 100 sage: a = random_matrix(ZZ,n) sage: v = vector(ZZ,n,range(n)) sage: x = a \ v sage: a * x == v True
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