For design documentation see sage.matrix.docs.
TESTS:
sage: A = Matrix(GF(5),3,3,srange(9))
sage: A == loads(dumps(A))
True
Returns a string defining a gap representation of self.
EXAMPLES:
sage: A = MatrixSpace(QQ,3,3)([0,1,2,3,4,5,6,7,8])
sage: g=gap(A) # indirect doctest
sage: g
[ [ 0, 1, 2 ], [ 3, 4, 5 ], [ 6, 7, 8 ] ]
sage: g.CharacteristicPolynomial()
x_1^3-12*x_1^2-18*x_1
sage: A = MatrixSpace(CyclotomicField(4),2,2)([0,1,2,3])
sage: g=gap(A)
sage: g
[ [ !0, !1 ], [ !2, !3 ] ]
sage: g.IsMatrix()
true
EXAMPLES:
sage: m = matrix(ZZ, [[1,2],[3,4]])
sage: macaulay2(m) #optional (indirect doctest)
| 1 2 |
| 3 4 |
sage: R.<x,y> = QQ[]
sage: m = matrix([[x,y],[1+x,1+y]])
sage: macaulay2(m) #optional
| x y |
| x+1 y+1 |
Return a string that evaluates in the given Magma session to this matrix.
EXAMPLES:
We first coerce a square matrix.
sage: A = MatrixSpace(QQ,3)([1,2,3,4/3,5/3,6/4,7,8,9])
sage: B = magma(A); B # (indirect doctest) optional - magma
[ 1 2 3]
[4/3 5/3 3/2]
[ 7 8 9]
sage: B.Type() # optional - magma
AlgMatElt
sage: B.Parent() # optional - magma
Full Matrix Algebra of degree 3 over Rational Field
We coerce a non-square matrix over
.
sage: A = MatrixSpace(Integers(8),2,3)([-1,2,3,4,4,-2])
sage: B = magma(A); B # optional - magma
[7 2 3]
[4 4 6]
sage: B.Type() # optional - magma
ModMatRngElt
sage: B.Parent() # optional - magma
Full RMatrixSpace of 2 by 3 matrices over IntegerRing(8)
sage: R.<x,y> = QQ[]
sage: A = MatrixSpace(R,2,2)([x+y,x-1,y+5,x*y])
sage: B = magma(A); B # optional - magma
[x + y x - 1]
[y + 5 x*y]
sage: R.<x,y> = ZZ[]
sage: A = MatrixSpace(R,2,2)([x+y,x-1,y+5,x*y])
sage: B = magma(A); B # optional - magma
[x + y x - 1]
[y + 5 x*y]
We coerce a matrix over a cyclotomic field, where the generator must be named during the coercion.
sage: K = CyclotomicField(9) ; z = K.0
sage: M = matrix(K,3,3,[0,1,3,z,z**4,z-1,z**17,1,0])
sage: M
[ 0 1 3]
[ zeta9 zeta9^4 zeta9 - 1]
[-zeta9^5 - zeta9^2 1 0]
sage: magma(M) # optional - magma
[ 0 1 3]
[ zeta9 zeta9^4 zeta9 - 1]
[-zeta9^5 - zeta9^2 1 0]
sage: magma(M**2) == magma(M)**2 # optional - magma
True
Return a Maple string representation of this matrix.
EXAMPLES:
sage: M = matrix(ZZ,2,range(4)) #optional
sage: maple(M) #optional (indirect doctest)
Matrix(2, 2, [[0,1],[2,3]])
sage: M = matrix(QQ,3,[1,2,3,4/3,5/3,6/4,7,8,9]) #optional
sage: maple(M) #optional
Matrix(3, 3, [[1,2,3],[4/3,5/3,3/2],[7,8,9]])
sage: P.<x> = ZZ[] #optional
sage: M = matrix(P, 2, [-9*x^2-2*x+2, x-1, x^2+8*x, -3*x^2+5]) #optional
sage: maple(M) #optional
Matrix(2, 2, [[-9*x^2-2*x+2,x-1],[x^2+8*x,-3*x^2+5]])
Return Mathematica string representation of this matrix.
EXAMPLES:
sage: A = MatrixSpace(QQ,3)([1,2,3,4/3,5/3,6/4,7,8,9])
sage: g = mathematica(A); g # optional
{{1, 2, 3}, {4/3, 5/3, 3/2}, {7, 8, 9}}
sage: A._mathematica_init_()
'{{1/1, 2/1, 3/1}, {4/3, 5/3, 3/2}, {7/1, 8/1, 9/1}}'
sage: A = matrix([[1,2],[3,4]])
sage: g = mathematica(A); g # optional
{{1, 2}, {3, 4}}
sage: a = matrix([[pi, sin(x)], [cos(x), 1/e]]); a
[ pi sin(x)]
[cos(x) e^(-1)]
sage: a._mathematica_init_()
'{{Pi, Sin[x]}, {Cos[x], Exp[-1]}}'
Return a string representation of this matrix in Maxima.
EXAMPLES:
sage: m = matrix(3,range(9)); m
[0 1 2]
[3 4 5]
[6 7 8]
sage: m._maxima_init_()
'matrix([0,1,2],[3,4,5],[6,7,8])'
sage: a = maxima(m); a
matrix([0,1,2],[3,4,5],[6,7,8])
sage: a.charpoly('x').expand()
-x^3+12*x^2+18*x
sage: m.charpoly()
x^3 - 12*x^2 - 18*x
Return the Pari matrix corresponding to self.
EXAMPLES:
sage: R.<x> = QQ['x']
sage: a = matrix(R,2,[x+1,2/3, x^2/2, 1+x^3]); a
[ x + 1 2/3]
[1/2*x^2 x^3 + 1]
sage: b = pari(a); b # indirect doctest
[x + 1, 2/3; 1/2*x^2, x^3 + 1]
sage: a.determinant()
x^4 + x^3 - 1/3*x^2 + x + 1
sage: b.matdet()
x^4 + x^3 - 1/3*x^2 + x + 1
This function preserves precision for entries of inexact type (e.g. reals):
sage: R = RealField(4) # 4 bits of precision
sage: a = matrix(R, 2, [1, 2, 3, 1]); a
[1.0 2.0]
[3.0 1.0]
sage: b = pari(a); b
[1.000000000, 2.000000000; 3.000000000, 1.000000000] # 32-bit
[1.00000000000000, 2.00000000000000; 3.00000000000000, 1.00000000000000] # 64-bit
sage: b[0][0].precision() # in words
3
Return a string defining a GP representation of self.
EXAMPLES:
sage: R.<x> = QQ['x']
sage: a = matrix(R,2,[x+1,2/3, x^2/2, 1+x^3]); a
[ x + 1 2/3]
[1/2*x^2 x^3 + 1]
sage: b = gp(a); b # indirect doctest
[x + 1, 2/3; 1/2*x^2, x^3 + 1]
sage: a.determinant()
x^4 + x^3 - 1/3*x^2 + x + 1
sage: b.matdet()
x^4 + x^3 - 1/3*x^2 + x + 1
Produce an expression which will reproduce this value when evaluated.
EXAMPLES:
sage: sage_input(matrix(QQ, 3, 3, [5..13])/7, verify=True)
# Verified
matrix(QQ, [[5/7, 6/7, 1], [8/7, 9/7, 10/7], [11/7, 12/7, 13/7]])
sage: sage_input(MatrixSpace(GF(5), 50, 50, sparse=True).random_element(density=0.002), verify=True)
# Verified
matrix(GF(5), 50, 50, {(7,43):4, (29,44):3, (35,4):4})
sage: from sage.misc.sage_input import SageInputBuilder
sage: matrix(RDF, [[3, 1], [4, 1]])._sage_input_(SageInputBuilder(), False)
{call: {atomic:matrix}({atomic:RDF}, {list: ({list: ({atomic:3}, {atomic:1})}, {list: ({atomic:4}, {atomic:1})})})}
sage: matrix(ZZ, 50, 50, {(9,17):1})._sage_input_(SageInputBuilder(), False)
{call: {atomic:matrix}({atomic:ZZ}, {atomic:50}, {atomic:50}, {dict: {{atomic:(9,17)}:{atomic:1}}})}
TESTS:
sage: sage_input(matrix(RR, 0, 3, []), verify=True)
# Verified
matrix(RR, 0, 3)
sage: sage_input(matrix(RR, 3, 0, []), verify=True)
# Verified
matrix(RR, 3, 0)
sage: sage_input(matrix(RR, 0, 0, []), verify=True)
# Verified
matrix(RR, 0, 0)
Creates a ScilabElement object based on self and returns it.
EXAMPLES:
sage: a = matrix([[1,2,3],[4,5,6],[7,8,9]]); a # optional - scilab [1 2 3] [4 5 6] [7 8 9] sage: b = scilab(a); b # optional - scilab (indirect doctest)
1. 2. 3. 4. 5. 6. 7. 8. 9.
AUTHORS:
Returns a string defining a Scilab representation of self.
EXAMPLES:
sage: a = matrix([[1,2,3],[4,5,6],[7,8,9]]); a # optional - scilab [1 2 3] [4 5 6] [7 8 9] sage: a._scilab_init_() # optional - scilab ‘[1,2,3;4,5,6;7,8,9]’
AUTHORS:
Returns the adjoint matrix of self (matrix of cofactors).
INPUT:
OUTPUT:
ALGORITHM: Use PARI
EXAMPLES:
sage: M = Matrix(ZZ,2,2,[5,2,3,4]) ; M
[5 2]
[3 4]
sage: N = M.adjoint() ; N
[ 4 -2]
[-3 5]
sage: M * N
[14 0]
[ 0 14]
sage: N * M
[14 0]
[ 0 14]
sage: M = Matrix(QQ,2,2,[5/3,2/56,33/13,41/10]) ; M
[ 5/3 1/28]
[33/13 41/10]
sage: N = M.adjoint() ; N
[ 41/10 -1/28]
[-33/13 5/3]
sage: M * N
[7363/1092 0]
[ 0 7363/1092]
TODO: Only implemented for matrices over ZZ or QQ PARI can deal with more general base rings
Return the augmented matrix of the form:
[self | other].
EXAMPLES:
sage: M = MatrixSpace(QQ,2,2)
sage: A = M([1,2, 3,4])
sage: A
[1 2]
[3 4]
sage: N = MatrixSpace(QQ,2,1)
sage: B = N([9,8])
sage: B
[9]
[8]
sage: A.augment(B)
[1 2 9]
[3 4 8]
sage: B.augment(A)
[9 1 2]
[8 3 4]
sage: M = MatrixSpace(QQ,3,4)
sage: A = M([1,2,3,4, 0,9,8,7, 2/3,3/4,4/5,9/8])
sage: A
[ 1 2 3 4]
[ 0 9 8 7]
[2/3 3/4 4/5 9/8]
sage: N = MatrixSpace(QQ,3,2)
sage: B = N([1,2, 3,4, 4,5])
sage: B
[1 2]
[3 4]
[4 5]
sage: A.augment(B)
[ 1 2 3 4 1 2]
[ 0 9 8 7 3 4]
[2/3 3/4 4/5 9/8 4 5]
sage: B.augment(A)
[ 1 2 1 2 3 4]
[ 3 4 0 9 8 7]
[ 4 5 2/3 3/4 4/5 9/8]
AUTHORS:
Return the block matrix that has self and other on the diagonal:
[ self 0 ]
[ 0 other ]
EXAMPLES:
sage: A = matrix(QQ[['t']], 2, range(1, 5))
sage: A.block_sum(100*A)
[ 1 2 0 0]
[ 3 4 0 0]
[ 0 0 100 200]
[ 0 0 300 400]
Return the i‘th column of this matrix as a vector.
This column is a dense vector if and only if the matrix is a dense matrix.
INPUT:
EXAMPLES:
sage: a = matrix(2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: a.column(1)
(1, 4)
If the column is negative, it wraps around, just like with list indexing, e.g., -1 gives the right-most column:
sage: a.column(-1)
(2, 5)
TESTS:
sage: a = matrix(2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: a.column(3)
...
IndexError: column index out of range
sage: a.column(-4)
...
IndexError: column index out of range
Return a list of the columns of self.
INPUT:
of columns which is safe to change.
If self is sparse, returns columns as sparse vectors, and if self is dense returns them as dense vectors.
EXAMPLES:
sage: matrix(3, [1..9]).columns()
[(1, 4, 7), (2, 5, 8), (3, 6, 9)]
sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).columns()
[(1.41421356237310, 2.71828182845905), (3.14159265358979, 0.000000000000000)]
sage: matrix(RR, 0, 2, []).columns()
[(), ()]
sage: matrix(RR, 2, 0, []).columns()
[]
sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)})
sage: parent(m.columns()[0])
Sparse vector space of dimension 3 over Real Field with 53 bits of precision
Return list of the dense columns of self.
INPUT:
EXAMPLES:
An example over the integers:
sage: a = matrix(3,3,range(9)); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: a.dense_columns()
[(0, 3, 6), (1, 4, 7), (2, 5, 8)]
We do an example over a polynomial ring:
sage: R.<x> = QQ[ ]
sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5]); a
[ x x^2]
[ 2/3*x x^5 + 1]
sage: a.dense_columns()
[(x, 2/3*x), (x^2, x^5 + 1)]
sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5], sparse=True)
sage: c = a.dense_columns(); c
[(x, 2/3*x), (x^2, x^5 + 1)]
sage: parent(c[1])
Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
If this matrix is sparse, return a dense matrix with the same entries. If this matrix is dense, return this matrix (not a copy).
Note
The definition of “dense” and “sparse” in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.
EXAMPLES:
sage: A = MatrixSpace(QQ,2, sparse=True)([1,2,0,1])
sage: A.is_sparse()
True
sage: B = A.dense_matrix()
sage: B.is_sparse()
False
sage: A*B
[1 4]
[0 1]
sage: A.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
In Sage, the product of a sparse and a dense matrix is always dense:
sage: (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
TESTS:
Make sure that subdivisions are preserved when switching between dense and sparse matrices:
sage: a = matrix(ZZ, 3, range(9))
sage: a.subdivide([1,2],2)
sage: a.get_subdivisions()
([1, 2], [2])
sage: b = a.sparse_matrix().dense_matrix()
sage: b.get_subdivisions()
([1, 2], [2])
Return list of the dense rows of self.
INPUT:
EXAMPLES:
sage: m = matrix(3, range(9)); m
[0 1 2]
[3 4 5]
[6 7 8]
sage: v = m.dense_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
sage: v is m.dense_rows()
False
sage: m.dense_rows(copy=False) is m.dense_rows(copy=False)
True
sage: m[0,0] = 10
sage: m.dense_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]
Return lift of self to the covering ring of the base ring R, which is by definition the ring returned by calling cover_ring() on R, or just R itself if the cover_ring method is not defined.
EXAMPLES:
sage: M = Matrix(Integers(7), 2, 2, [5, 9, 13, 15]) ; M
[5 2]
[6 1]
sage: M.lift()
[5 2]
[6 1]
sage: parent(M.lift())
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
The field QQ doesn’t have a cover_ring method:
sage: hasattr(QQ, 'cover_ring')
False
So lifting a matrix over QQ gives back the same exact matrix.
sage: B = matrix(QQ, 2, [1..4])
sage: B.lift()
[1 2]
[3 4]
sage: B.lift() is B
True
Return the matrix constructed from self using columns with indices in the columns list.
EXAMPLES:
sage: M = MatrixSpace(Integers(8),3,3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_columns([2,1])
[2 1]
[5 4]
[0 7]
Return the matrix constructed from self using rows with indices in the rows list.
EXAMPLES:
sage: M = MatrixSpace(Integers(8),3,3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_rows([2,1])
[6 7 0]
[3 4 5]
Return the matrix constructed from self from the given rows and columns.
EXAMPLES:
sage: M = MatrixSpace(Integers(8),3,3)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 0]
sage: A.matrix_from_rows_and_columns([1], [0,2])
[3 5]
sage: A.matrix_from_rows_and_columns([1,2], [1,2])
[4 5]
[7 0]
Note that row and column indices can be reordered or repeated:
sage: A.matrix_from_rows_and_columns([2,1], [2,1])
[0 7]
[5 4]
For example here we take from row 1 columns 2 then 0 twice, and do this 3 times.
sage: A.matrix_from_rows_and_columns([1,1,1],[2,0,0])
[5 3 3]
[5 3 3]
[5 3 3]
AUTHORS:
Return copy of this matrix, but with entries viewed as elements of the fraction field of the base ring (assuming it is defined).
EXAMPLES:
sage: A = MatrixSpace(IntegerRing(),2)([1,2,3,4])
sage: B = A.matrix_over_field()
sage: B
[1 2]
[3 4]
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
Return the ambient matrix space of self.
INPUT:
EXAMPLES:
sage: m = matrix(3, [1..9])
sage: m.matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
sage: m.matrix_space(ncols=2)
Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
sage: m.matrix_space(1)
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m.matrix_space(1, 2, True)
Full MatrixSpace of 1 by 2 sparse matrices over Integer Ring
Create a matrix in the parent of this matrix with the given number of rows, columns, etc. The default parameters are the same as for self.
INPUT:
These three variables get sent to matrix_space():
The remaining three variables (coerce, entries, and copy) are used by sage.matrix.matrix_space.MatrixSpace() to construct the new matrix.
Warning
This function called with no arguments returns the zero matrix of the same dimension and sparseness of self.
EXAMPLES:
sage: A = matrix(ZZ,2,2,[1,2,3,4]); A [1 2] [3 4] sage: A.new_matrix() [0 0] [0 0] sage: A.new_matrix(1,1) [0] sage: A.new_matrix(3,3).parent() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
sage: A = matrix(RR,2,3,[1.1,2.2,3.3,4.4,5.5,6.6]); A
[1.10000000000000 2.20000000000000 3.30000000000000]
[4.40000000000000 5.50000000000000 6.60000000000000]
sage: A.new_matrix()
[0.000000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000 0.000000000000000]
sage: A.new_matrix().parent()
Full MatrixSpace of 2 by 3 dense matrices over Real Field with 53 bits of precision
Return the Numpy matrix associated to this matrix.
INPUT:
EXAMPLES:
sage: a = matrix(3,range(12))
sage: a.numpy()
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])
sage: a.numpy('f')
array([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.]], dtype=float32)
sage: a.numpy('d')
array([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.]])
sage: a.numpy('B')
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]], dtype=uint8)
Type numpy.typecodes for a list of the possible typecodes:
sage: import numpy
sage: sorted(numpy.typecodes.items())
[('All', '?bhilqpBHILQPfdgFDGSUVO'), ('AllFloat', 'fdgFDG'), ('AllInteger', 'bBhHiIlLqQpP'), ('Character', 'c'), ('Complex', 'FDG'), ('Float', 'fdg'), ('Integer', 'bhilqp'), ('UnsignedInteger', 'BHILQP')]
Return the i‘th row of this matrix as a vector.
This row is a dense vector if and only if the matrix is a dense matrix.
INPUT:
EXAMPLES:
sage: a = matrix(2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: a.row(0)
(0, 1, 2)
sage: a.row(1)
(3, 4, 5)
sage: a.row(-1) # last row
(3, 4, 5)
TESTS:
sage: a = matrix(2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: a.row(2)
...
IndexError: row index out of range
sage: a.row(-3)
...
IndexError: row index out of range
Return a list of the rows of self.
INPUT:
If self is sparse, returns rows as sparse vectors, and if self is dense returns them as dense vectors.
EXAMPLES:
sage: matrix(3, [1..9]).rows()
[(1, 2, 3), (4, 5, 6), (7, 8, 9)]
sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).rows()
[(1.41421356237310, 3.14159265358979), (2.71828182845905, 0.000000000000000)]
sage: matrix(RR, 0, 2, []).rows()
[]
sage: matrix(RR, 2, 0, []).rows()
[(), ()]
sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)})
sage: parent(m.rows()[0])
Sparse vector space of dimension 3 over Real Field with 53 bits of precision
Sets the entries of column col in self to be the entries of v.
EXAMPLES:
sage: A = matrix([[1,2],[3,4]]); A
[1 2]
[3 4]
sage: A.set_column(0, [0,0]); A
[0 2]
[0 4]
sage: A.set_column(1, [0,0]); A
[0 0]
[0 0]
sage: A.set_column(2, [0,0]); A
...
IndexError: index out of range
sage: A.set_column(0, [0,0,0])
...
ValueError: v must be of length 2
Sets the entries of row row in self to be the entries of v.
EXAMPLES:
sage: A = matrix([[1,2],[3,4]]); A
[1 2]
[3 4]
sage: A.set_row(0, [0,0]); A
[0 0]
[3 4]
sage: A.set_row(1, [0,0]); A
[0 0]
[0 0]
sage: A.set_row(2, [0,0]); A
...
IndexError: index out of range
sage: A.set_row(0, [0,0,0])
...
ValueError: v must be of length 2
Return list of the sparse columns of self.
INPUT:
modify it safely
EXAMPLES:
sage: a = matrix(2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: v = a.sparse_columns(); v
[(0, 3), (1, 4), (2, 5)]
sage: v[1].is_sparse()
True
If this matrix is dense, return a sparse matrix with the same entries. If this matrix is sparse, return this matrix (not a copy).
Note
The definition of “dense” and “sparse” in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.
EXAMPLES:
sage: A = MatrixSpace(QQ,2, sparse=False)([1,2,0,1])
sage: A.is_sparse()
False
sage: B = A.sparse_matrix()
sage: B.is_sparse()
True
sage: A
[1 2]
[0 1]
sage: B
[1 2]
[0 1]
sage: A*B
[1 4]
[0 1]
sage: A.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: B.parent()
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: (A*B).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: (B*A).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
Return list of the sparse rows of self.
INPUT:
modify it safely
EXAMPLES:
sage: m = Mat(ZZ,3,3,sparse=True)(range(9)); m
[0 1 2]
[3 4 5]
[6 7 8]
sage: v = m.sparse_rows(); v
[(0, 1, 2), (3, 4, 5), (6, 7, 8)]
sage: m.sparse_rows(copy=False) is m.sparse_rows(copy=False)
True
sage: v[1].is_sparse()
True
sage: m[0,0] = 10
sage: m.sparse_rows()
[(10, 1, 2), (3, 4, 5), (6, 7, 8)]
Return the augmented matrix self on top of other:
[ self ]
[ other ]
EXAMPLES:
sage: M = Matrix(QQ, 2, 3, range(6))
sage: N = Matrix(QQ, 1, 3, [10,11,12])
sage: M.stack(N)
[ 0 1 2]
[ 3 4 5]
[10 11 12]
Return the matrix constructed from self using the specified range of rows and columns.
INPUT:
SEE ALSO:
The functions matrix_from_rows(), matrix_from_columns(), and matrix_from_rows_and_columns() allow one to select arbitrary subsets of rows and/or columns.
EXAMPLES:
Take the submatrix starting from entry (1,1) in a
matrix:
sage: m = matrix(4, [1..16])
sage: m.submatrix(1, 1)
[ 6 7 8]
[10 11 12]
[14 15 16]
Same thing, except take only two rows:
sage: m.submatrix(1, 1, 2)
[ 6 7 8]
[10 11 12]
And now take only one column:
sage: m.submatrix(1, 1, 2, 1)
[ 6]
[10]
You can take zero rows or columns if you want:
sage: m.submatrix(1, 1, 0)
[]
sage: parent(m.submatrix(1, 1, 0))
Full MatrixSpace of 0 by 3 dense matrices over Integer Ring