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31.6 Base class for matrices, part 1

Module: sage.matrix.matrix1

File: sage/matrix/matrix1.pyx (starting at line 1)

Base class for matrices, part 1

For design documentation see matrix/docs.py.

Class: Matrix

class Matrix

Functions: adjoint,$ $ augment,$ $ block_sum,$ $ column,$ $ columns,$ $ dense_columns,$ $ dense_matrix,$ $ dense_rows,$ $ lift,$ $ matrix_from_columns,$ $ matrix_from_rows,$ $ matrix_from_rows_and_columns,$ $ matrix_over_field,$ $ matrix_space,$ $ new_matrix,$ $ numeric_array,$ $ numpy,$ $ row,$ $ rows,$ $ sparse_columns,$ $ sparse_matrix,$ $ sparse_rows,$ $ stack,$ $ submatrix

adjoint( )

Returns the adjoint matrix of self (matrix of cofactors). 

INPUT:
    M -- a square matrix

OUTPUT:
    N -- the adjoint matrix, such that
        N * M = M * N = M.parent(M.det())

ALGORITHM: Use PARI 

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

augment( )

Return the augmented matrix of the form [self | other]. 

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]

Author: Naqi Jaffery (2006-01-24): examples

block_sum( )

Return the block matrix that has self and other on the diagonal: [self | 0 ] [ 0 | other ] 

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]

column( )

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:
    i -- integer
    from_list -- bool (default: False); if true, returns the
                 ith element of self.columns(), which may be
                 faster, but requires building a list of all
                 columns the first time it is called after an
                 entry of the matrix is changed.


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)

columns( )

Return a list of the columns of self. 

INPUT:
    copy -- (default: True) if True, return a copy of the list 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.

dense_columns( )

Return list of the dense columns of self. 

INPUT:
    copy -- (default: True) if True, return a copy so you can modify it
safely

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

dense_matrix( )

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. 

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

dense_rows( )

Return list of the dense rows of self. 

INPUT:
    copy -- (default: True) if True, return a copy so you can modify it
safely


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)]

lift( )

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

matrix_from_columns( )

Return the matrix constructed from self using columns with indices in the columns list. 

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]

matrix_from_rows( )

Return the matrix constructed from self using rows with indices in the rows list. 

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]

matrix_from_rows_and_columns( )

Return the matrix constructed from self from the given rows and columns. 

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]

Author: Jaap Spies (2006-02-18)

matrix_over_field( )

Return copy of this matrix, but with entries viewed as elements of the fraction field of the base ring (assuming it is defined). 

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

new_matrix( )

Create a matrix in the parent of this space with the given number of rows, columns, etc. The default parameters are the same as for self.

WARNING: This function called with no arguments returns the 0 matrix by default.

numeric_array( )

Return the Numeric array associated to this matrix (if possible). All entries must be coercible to the given typecode. 

INPUT:
    typecode -- optional (default: Numeric.Float64)

numpy( )

Return the Numpy matrix associated to this matrix. 

INPUT:
    dtype  - The desired data-type for the array.  If not given, then
             the type will be determined as the minimum type required
             to hold the objects in the sequence.


sage: a = matrix(3,range(12))
sage: a.numpy()
array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11]], dtype=object)
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: numpy.typecodes
{'All': '?bhilqpBHILQPfdgFDGSUVO', 'AllInteger': 'bBhHiIlLqQpP',
'AllFloat': 'fdgFDG', 'UnsignedInteger': 'BHILQP', 'Float': 'fdg',
'Character': 'S1', 'Complex': 'FDG', 'Integer': 'bhilqp'}

row( )

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:
    i -- integer
    from_list -- bool (default: False); if true, returns the
                 ith element of self.rows(), which may be
                 faster, but requires building a list of all
                 rows the first time it is called after an
                 entry of the matrix is changed.


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)

rows( )

Return a list of the rows of self. 

INPUT:
    copy -- (default: True) if True, return a copy of the list of rows,
which is safe to change.

If self is sparse, returns rows as sparse vectors, and if self is dense returns them as dense vectors.

sparse_columns( )

Return list of the sparse columns of self. 

INPUT:
     copy -- (default: True) if True, return a copy so you can modify it
safely


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

sparse_matrix( )

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. 

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

sparse_rows( )

Return list of the sparse rows of self. 

INPUT:
    copy -- (default: True) if True, return a copy so you can modify it
safely


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)]

stack( )

Return the augmented matrix self on top of other: [ self ] [ other ] 

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]

Special Functions: _gap_init_,$ $ _magma_init_,$ $ _mathematica_init_,$ $ _maxima_init_,$ $ _pari_init_,$ $ _singular_

_gap_init_( )

sage: A = MatrixSpace(QQ,3)([1,2,3,4/3,5/3,6/4,7,8,9])
sage: g = gap(A); g
[ [ 1, 2, 3 ], [ 4/3, 5/3, 3/2 ], [ 7, 8, 9 ] ]
sage: g.IsMatrix()
true

_magma_init_( )

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                       # optional
[  1   2   3]
[4/3 5/3 3/2]
[  7   8   9]
sage: B.Type()                              # optional
AlgMatElt
sage: B.Parent()                            # optional
Full Matrix Algebra of degree 3 over Rational Field

We coerce a non-square matrix over $ \mathbf{Z}/8\mathbf{Z}$. 

sage: A = MatrixSpace(Integers(8),2,3)([-1,2,3,4,4,-2])   
sage: B = magma(A); B                       # optional
[7 2 3]
[4 4 6]
sage: B.Type()                              # optional
ModMatRngElt
sage: B.Parent()                            # optional
Full RMatrixSpace of 2 by 3 matrices over IntegerRing(8)

_mathematica_init_( )

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}}

_maxima_init_( )

Return string representation of this matrix in maxima. 

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

_pari_init_( )

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
[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

_singular_( )

Tries to coerce this matrix to a singular matrix.

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