AUTHORS:
EXAMPLES:
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: TestSuite(a).run()
sage: Matrix(ZZ,0,0).inverse()
[]
Bases: sage.matrix.matrix_dense.Matrix_dense
Matrix over the integers, implemented using FLINT.
On a 32-bit machine, they can have at most \(2^{32}-1\) rows or columns. On a 64-bit machine, matrices can have at most \(2^{64}-1\) rows or columns.
EXAMPLES:
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
Block Korkin-Zolotarev reduction.
INPUT:
NLT SPECIFIC INPUTS:
fpLLL SPECIFIC INPUTS:
EXAMPLES:
sage: A = Matrix(ZZ,3,3,range(1,10))
sage: A.BKZ()
[ 0 0 0]
[ 2 1 0]
[-1 1 3]
sage: A = Matrix(ZZ,3,3,range(1,10))
sage: A.BKZ(use_givens=True)
[ 0 0 0]
[ 2 1 0]
[-1 1 3]
sage: A = Matrix(ZZ,3,3,range(1,10))
sage: A.BKZ(fp="fp")
[ 0 0 0]
[ 2 1 0]
[-1 1 3]
ALGORITHM:
Calls either NTL or fpLLL.
REFERENCES:
[SH95] | C. P. Schnorr and H. H. Hörner. Attacking the Chor-Rivest Cryptosystem by Improved Lattice Reduction. Advances in Cryptology - EUROCRYPT ‘95. LNCS Volume 921, 1995, pp 1-12. |
[GL96] | G. Golub and C. van Loan. Matrix Computations. 3rd edition, Johns Hopkins Univ. Press, 1996. |
Return LLL reduced or approximated LLL reduced lattice \(R\) for this matrix interpreted as a lattice.
A lattice \((b_1, b_2, ..., b_d)\) is \((\delta, \eta)\)-LLL-reduced if the two following conditions hold:
where \(μ_{i,j} = \langle b_i, b_j^* \rangle / \langle b_j^*, b_j^* \rangle\) and \(b_i^*\) is the \(i\)-th vector of the Gram-Schmidt orthogonalisation of \((b_1, b_2, ..., b_d)\).
The default reduction parameters are \(\delta = 3/4\) and \(\eta = 0.501\). The parameters \(\delta\) and \(\eta\) must satisfy: \(0.25 < \delta \leq 1.0\) and \(0.5 \leq \eta < \sqrt{\delta}\). Polynomial time complexity is only guaranteed for \(\delta < 1\). Not every algorithm admits the case \(\delta = 1\).
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 at least \(2\), some more verbose output is printed during the computation.
AVAILABLE ALGORITHMS:
OUTPUT:
A matrix over the integers.
EXAMPLES:
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
The case \(\delta = 1\) is not always supported:
sage: L = X.LLL(delta=2)
Traceback (most recent call last):
...
TypeError: delta must be <= 1
sage: L = X.LLL(delta=1) # not tested, will eat lots of ram
Traceback (most recent call last):
...
RuntimeError: infinite loop in LLL
sage: L = X.LLL(delta=1, algorithm='NTL:LLL')
sage: L[-1]
(-100, -3, -1, 13, -1, -4, 2, 3, 4, 5, -1)
TESTS:
sage: matrix(ZZ, 0, 0).LLL()
[]
sage: matrix(ZZ, 3, 0).LLL()
[]
sage: matrix(ZZ, 0, 3).LLL()
[]
sage: M = matrix(ZZ, [[1,2,3],[31,41,51],[101,201,301]])
sage: A = M.LLL()
sage: A
[ 0 0 0]
[-1 0 1]
[ 1 1 1]
sage: B = M.LLL(algorithm='NTL:LLL')
sage: C = M.LLL(algorithm='NTL:LLL', fp=None)
sage: D = M.LLL(algorithm='NTL:LLL', fp='fp')
sage: F = M.LLL(algorithm='NTL:LLL', fp='xd')
sage: G = M.LLL(algorithm='NTL:LLL', fp='rr')
sage: A == B == C == D == F == G
True
sage: H = M.LLL(algorithm='NTL:LLL', fp='qd')
Traceback (most recent call last):
...
TypeError: algorithm NTL:LLL_QD not supported
Note
See ntl.mat_ZZ or sage.libs.fplll.fplll for details on the used algorithms.
LLL reduction of the lattice whose gram matrix is self.
INPUT:
OUTPUT:
U - unimodular transformation matrix such that U.T * M * U is LLL-reduced.
ALGORITHM: Use PARI
EXAMPLES:
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 no longer raise a ValueError:
sage: Matrix(ZZ,2,2,[2,6,6,3]).LLL_gram()
[-3 -1]
[ 1 0]
sage: Matrix(ZZ,2,2,[1,0,0,-1]).LLL_gram()
[ 0 -1]
[ 1 0]
Returns the antitranspose of self, without changing self.
EXAMPLES:
sage: A = matrix(2,3,range(6))
sage: type(A)
<type 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: A.antitranspose()
[5 2]
[4 1]
[3 0]
sage: A
[0 1 2]
[3 4 5]
sage: A.subdivide(1,2); A
[0 1|2]
[---+-]
[3 4|5]
sage: A.antitranspose()
[5|2]
[-+-]
[4|1]
[3|0]
Returns a new matrix formed by appending the matrix (or vector) right on the right side of self.
INPUT:
OUTPUT:
A new matrix formed by appending right onto the right side of self. If right is a vector (or free module element) then in this context it is appropriate to consider it as a column vector. (The code first converts a vector to a 1-column matrix.)
EXAMPLES:
sage: A = matrix(ZZ, 4, 5, range(20))
sage: B = matrix(ZZ, 4, 3, range(12))
sage: A.augment(B)
[ 0 1 2 3 4 0 1 2]
[ 5 6 7 8 9 3 4 5]
[10 11 12 13 14 6 7 8]
[15 16 17 18 19 9 10 11]
A vector may be augmented to a matrix.
sage: A = matrix(ZZ, 3, 5, range(15))
sage: v = vector(ZZ, 3, range(3))
sage: A.augment(v)
[ 0 1 2 3 4 0]
[ 5 6 7 8 9 1]
[10 11 12 13 14 2]
The subdivide option will add a natural subdivision between self and right. For more details about how subdivisions are managed when augmenting, see sage.matrix.matrix1.Matrix.augment().
sage: A = matrix(ZZ, 3, 5, range(15))
sage: B = matrix(ZZ, 3, 3, range(9))
sage: A.augment(B, subdivide=True)
[ 0 1 2 3 4| 0 1 2]
[ 5 6 7 8 9| 3 4 5]
[10 11 12 13 14| 6 7 8]
Errors are raised if the sizes are incompatible.
sage: A = matrix(ZZ, [[1, 2],[3, 4]])
sage: B = matrix(ZZ, [[10, 20], [30, 40], [50, 60]])
sage: A.augment(B)
Traceback (most recent call last):
...
TypeError: number of rows must be the same, not 2 != 3
INPUT:
Note
Linbox charpoly disabled on 64-bit machines, since it hangs in many cases.
EXAMPLES:
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()
x^3 - 3990*x^2 - 266000*x
TESTS:
The cached polynomial should be independent of the var argument (trac ticket #12292). We check (indirectly) that the second call uses the cached value by noting that its result is not cached:
sage: M = MatrixSpace(ZZ, 2)
sage: A = M(range(0, 2^2))
sage: type(A)
<type 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: A.charpoly('x')
x^2 - 3*x - 2
sage: A.charpoly('y')
y^2 - 3*y - 2
sage: A._cache['charpoly_linbox']
x^2 - 3*x - 2
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:
EXAMPLES:
sage: t = ModularSymbols(11,sign=1).hecke_matrix(2)
sage: w = t.change_ring(ZZ)
sage: w.list()
[3, -1, 0, -2]
Return the determinant of this matrix.
INPUT:
algorithm
proof - bool or None; if None use proof.linear_algebra(); only relevant for the padic algorithm.
Note
It would be VERY VERY hard for det to fail even with proof=False.
stabilize - if proof is False, require det to be the same for this many CRT primes in a row. Ignored if proof is True.
ALGORITHM: The p-adic algorithm works by first finding a random vector v, then solving A*x = v and taking the denominator \(d\). This gives a divisor of the determinant. Then we compute \(\det(A)/d\) using a multimodular algorithm and the Hadamard bound, skipping primes that divide \(d\).
EXAMPLES:
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
We have a special-case algorithm for 4 x 4 determinants:
sage: A = matrix(ZZ,4,[1,2,3,4,4,3,2,1,0,5,0,1,9,1,2,3])
sage: A.determinant()
270
Next we try the Linbox det. Note that we must have proof=False.
sage: A = matrix(ZZ,5,[1,2,3,4,5,4,6,3,2,1,7,9,7,5,2,1,4,6,7,8,3,2,4,6,7])
sage: A.determinant(algorithm='linbox')
Traceback (most recent call last):
...
RuntimeError: you must pass the proof=False option to the determinant command to use LinBox's det algorithm
sage: A.determinant(algorithm='linbox',proof=False)
-21
sage: A._clear_cache()
sage: A.determinant()
-21
A bigger example:
sage: A = random_matrix(ZZ,30)
sage: d = A.determinant()
sage: A._clear_cache()
sage: A.determinant(algorithm='linbox',proof=False) == d
True
TESTS:
This shows that we can compute determinants for all sizes up to 80. The check that the determinant of a squared matrix is a square is a sanity check that the result is probably correct:
sage: for s in [1..80]: # long time
....: M = random_matrix(ZZ, s)
....: d = (M*M).determinant()
....: assert d.is_square()
Return the echelon form of this matrix over the integers, also known as the hermite normal form (HNF).
INPUT:
OUTPUT:
The Hermite normal form (=echelon form over \(\ZZ\)) of self.
EXAMPLES:
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]
Getting a transformation matrix in the nonsquare case:
sage: A = matrix(ZZ,5,3,[1..15])
sage: H, U = A.hermite_form(transformation=True, include_zero_rows=False)
sage: H
[1 2 3]
[0 3 6]
sage: U
[ 0 0 0 4 -3]
[ 0 0 0 13 -10]
sage: U*A == H
True
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]
Illustrate using various algorithms.:
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari0')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari4')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='padic')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='default')
[1 2 3]
[0 3 6]
[0 0 0]
The ‘ntl’ algorithm doesn’t work on matrices that do not have full rank.:
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='ntl')
Traceback (most recent call last):
...
ValueError: ntl only computes HNF for square matrices of full rank.
sage: matrix(ZZ,3,[0] +[2..9]).hermite_form(algorithm='ntl')
[1 0 0]
[0 1 0]
[0 0 3]
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]
Check that #12280 is fixed:
sage: m = matrix([(-2, 1, 9, 2, -8, 1, -3, -1, -4, -1),
... (5, -2, 0, 1, 0, 4, -1, 1, -2, 0),
... (-11, 3, 1, 0, -3, -2, -1, -11, 2, -2),
... (-1, 1, -1, -2, 1, -1, -1, -1, -1, 7),
... (-2, -1, -1, 1, 1, -2, 1, 0, 2, -4)]).stack(
... 200 * identity_matrix(ZZ, 10))
sage: matrix(ZZ,m).hermite_form(algorithm='pari', include_zero_rows=False)
[ 1 0 2 0 13 5 1 166 72 69]
[ 0 1 1 0 20 4 15 195 65 190]
[ 0 0 4 0 24 5 23 22 51 123]
[ 0 0 0 1 23 7 20 105 60 151]
[ 0 0 0 0 40 4 0 80 36 68]
[ 0 0 0 0 0 10 0 100 190 170]
[ 0 0 0 0 0 0 25 0 100 150]
[ 0 0 0 0 0 0 0 200 0 0]
[ 0 0 0 0 0 0 0 0 200 0]
[ 0 0 0 0 0 0 0 0 0 200]
sage: matrix(ZZ,m).hermite_form(algorithm='padic', include_zero_rows=False)
[ 1 0 2 0 13 5 1 166 72 69]
[ 0 1 1 0 20 4 15 195 65 190]
[ 0 0 4 0 24 5 23 22 51 123]
[ 0 0 0 1 23 7 20 105 60 151]
[ 0 0 0 0 40 4 0 80 36 68]
[ 0 0 0 0 0 10 0 100 190 170]
[ 0 0 0 0 0 0 25 0 100 150]
[ 0 0 0 0 0 0 0 200 0 0]
[ 0 0 0 0 0 0 0 0 200 0]
[ 0 0 0 0 0 0 0 0 0 200]
Return the elementary divisors of self, in order.
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 left multiplication of this matrix. They are ordered in reverse by divisibility.
INPUT:
OUTPUT: list of integers
Note
These are the invariants of the cokernel of left multiplication:
sage: M = Matrix([[3,0,1],[0,1,0]])
sage: M
[3 0 1]
[0 1 0]
sage: M.elementary_divisors()
[1, 1]
sage: M.transpose().elementary_divisors()
[1, 1, 0]
EXAMPLES:
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
Return the Frobenius form (rational canonical form) of this matrix.
INPUT:
flag – 0 (default), 1 or 2 as follows:
- 0 – (default) return the Frobenius form of this matrix.
- 1 – return only the elementary divisor polynomials, as polynomials in var.
- 2 – return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that \(M=B^{-1}FB\).
var – a string (default: ‘x’)
ALGORITHM: uses PARI’s matfrobenius()
EXAMPLES:
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: F, B = A.frobenius(2)
sage: A == B^(-1)*F*B
True
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.
AUTHORS:
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.
Return the gcd of all entries of self; very fast.
EXAMPLES:
sage: a = matrix(ZZ,2, [6,15,-6,150])
sage: a.gcd()
3
Return the height of this matrix, i.e., the max absolute value of the entries of the matrix.
OUTPUT: A nonnegative integer.
EXAMPLE:
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
Return the echelon form of this matrix over the integers, also known as the hermite normal form (HNF).
INPUT:
OUTPUT:
The Hermite normal form (=echelon form over \(\ZZ\)) of self.
EXAMPLES:
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]
Getting a transformation matrix in the nonsquare case:
sage: A = matrix(ZZ,5,3,[1..15])
sage: H, U = A.hermite_form(transformation=True, include_zero_rows=False)
sage: H
[1 2 3]
[0 3 6]
sage: U
[ 0 0 0 4 -3]
[ 0 0 0 13 -10]
sage: U*A == H
True
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]
Illustrate using various algorithms.:
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari0')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='pari4')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='padic')
[1 2 3]
[0 3 6]
[0 0 0]
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='default')
[1 2 3]
[0 3 6]
[0 0 0]
The ‘ntl’ algorithm doesn’t work on matrices that do not have full rank.:
sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm='ntl')
Traceback (most recent call last):
...
ValueError: ntl only computes HNF for square matrices of full rank.
sage: matrix(ZZ,3,[0] +[2..9]).hermite_form(algorithm='ntl')
[1 0 0]
[0 1 0]
[0 0 3]
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]
Check that #12280 is fixed:
sage: m = matrix([(-2, 1, 9, 2, -8, 1, -3, -1, -4, -1),
... (5, -2, 0, 1, 0, 4, -1, 1, -2, 0),
... (-11, 3, 1, 0, -3, -2, -1, -11, 2, -2),
... (-1, 1, -1, -2, 1, -1, -1, -1, -1, 7),
... (-2, -1, -1, 1, 1, -2, 1, 0, 2, -4)]).stack(
... 200 * identity_matrix(ZZ, 10))
sage: matrix(ZZ,m).hermite_form(algorithm='pari', include_zero_rows=False)
[ 1 0 2 0 13 5 1 166 72 69]
[ 0 1 1 0 20 4 15 195 65 190]
[ 0 0 4 0 24 5 23 22 51 123]
[ 0 0 0 1 23 7 20 105 60 151]
[ 0 0 0 0 40 4 0 80 36 68]
[ 0 0 0 0 0 10 0 100 190 170]
[ 0 0 0 0 0 0 25 0 100 150]
[ 0 0 0 0 0 0 0 200 0 0]
[ 0 0 0 0 0 0 0 0 200 0]
[ 0 0 0 0 0 0 0 0 0 200]
sage: matrix(ZZ,m).hermite_form(algorithm='padic', include_zero_rows=False)
[ 1 0 2 0 13 5 1 166 72 69]
[ 0 1 1 0 20 4 15 195 65 190]
[ 0 0 4 0 24 5 23 22 51 123]
[ 0 0 0 1 23 7 20 105 60 151]
[ 0 0 0 0 40 4 0 80 36 68]
[ 0 0 0 0 0 10 0 100 190 170]
[ 0 0 0 0 0 0 25 0 100 150]
[ 0 0 0 0 0 0 0 200 0 0]
[ 0 0 0 0 0 0 0 0 200 0]
[ 0 0 0 0 0 0 0 0 0 200]
Return the index of self in its saturation.
INPUT:
OUTPUT:
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.
EXAMPLES:
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]
Create a new matrix from self with.
INPUT:
EXAMPLES:
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]
Return True if this lattice is \((\delta, \eta)\)-LLL reduced. See self.LLL for a definition of LLL reduction.
INPUT:
EXAMPLES:
sage: A = random_matrix(ZZ, 10, 10)
sage: L = A.LLL()
sage: A.is_LLL_reduced()
False
sage: L.is_LLL_reduced()
True
INPUT:
Note
Linbox charpoly disabled on 64-bit machines, since it hangs in many cases.
EXAMPLES:
sage: A = matrix(ZZ,6, range(36))
sage: A.minpoly()
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()
x^4 - 2695*x^3 - 257964*x^2 + 1693440*x
Return the pivot column positions of this matrix.
OUTPUT: a tuple of Python integers: the position of the first nonzero entry in each row of the echelon form.
EXAMPLES:
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]
Return the product of the sums of the entries in the submatrix of self with given columns.
INPUT:
OUTPUT: an integer
EXAMPLES:
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
Randomize density proportion of the entries of this matrix, leaving the rest unchanged.
The parameters are the same as the ones for the integer ring’s random_element function.
If x and y are given, randomized entries of this matrix have to be between x and y and have density 1.
INPUT:
OUTPUT:
EXAMPLES:
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]
Return the rank of this matrix.
OUTPUT:
Note
The rank is cached.
ALGORITHM: If set to 'modp', first check if the matrix has maximum possible rank by working modulo one random prime. If not call LinBox’s rank function.
EXAMPLES:
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:
OUTPUT:
EXAMPLES: 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 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
TEST:
Check that trac:\(9345\) is fixed:
sage: A = random_matrix(ZZ, 3, 3)
sage: A.rational_reconstruction(0)
Traceback (most recent call last):
...
ZeroDivisionError: The modulus cannot be zero
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 \(\ZZ\) module \(M\) embedded in \(\ZZ^n\) is the a module \(S\) that contains \(M\) with finite index such that \(\ZZ^n/S\) is torsion free. This function takes the row span \(M\) of self, and finds another matrix of full rank with row span the saturation of \(M\).
INPUT:
OUTPUT:
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 \(hnf(transpose(A))^{-1}*A\) is a saturation of A.
EXAMPLES:
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.
EXAMPLES:
sage: A = MatrixSpace(IntegerRing(), 3)(range(9))
sage: D, U, V = A.smith_form()
sage: D
[1 0 0]
[0 3 0]
[0 0 0]
sage: U
[ 0 1 0]
[ 0 -1 1]
[-1 2 -1]
sage: V
[-1 4 1]
[ 1 -3 -2]
[ 0 0 1]
sage: U*A*V
[1 0 0]
[0 3 0]
[0 0 0]
It also makes sense for nonsquare matrices:
sage: A = Matrix(ZZ,3,2,range(6))
sage: D, U, V = A.smith_form()
sage: D
[1 0]
[0 2]
[0 0]
sage: U
[ 0 1 0]
[ 0 -1 1]
[-1 2 -1]
sage: V
[-1 3]
[ 1 -2]
sage: U * A * V
[1 0]
[0 2]
[0 0]
Empty matrices are handled sensibly (see trac #3068):
sage: m = MatrixSpace(ZZ, 2,0)(0); d,u,v = m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(ZZ, 0,2)(0); d,u,v = m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(ZZ, 0,0)(0); d,u,v = m.smith_form(); u*m*v == d
True
See also
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 \(M = -M^t\). Alternating means that the diagonal of \(M\) is identically zero.
A symplectic basis is a basis of the form \(e_1, \ldots, e_j, f_1, \ldots f_j, z_1, \dots, z_k\) such that
\(z_i M v^t\) = 0 for all vectors \(v\)
\(e_i M {e_j}^t = 0\) for all \(i, j\)
\(f_i M {f_j}^t = 0\) for all \(i, j\)
\(e_i M {f_i}^t = 1\) for all \(i\)
\(j\).
The ordering for the factors \(d_{i} | d_{i+1}\) 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.
EXAMPLES:
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 1 0 0]
[ 0 0 0 2 0]
[-1 0 0 0 0]
[ 0 -2 0 0 0]
[ 0 0 0 0 0]
sage: F == C * E * C.transpose()
True
sage: E.smith_form()[0]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 0]
Returns the transpose of self, without changing self.
EXAMPLES:
We create a matrix, compute its transpose, and note that the original matrix is not changed.
sage: A = matrix(ZZ,2,3,xrange(6))
sage: type(A)
<type 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
sage: B = A.transpose()
sage: print B
[0 3]
[1 4]
[2 5]
sage: print A
[0 1 2]
[3 4 5]
.T is a convenient shortcut for the transpose:
sage: A.T
[0 3]
[1 4]
[2 5]
sage: A.subdivide(None, 1); A
[0|1 2]
[3|4 5]
sage: A.transpose()
[0 3]
[---]
[1 4]
[2 5]