Linear algebra

Vector spaces

The VectorSpace command creates a vector space class, from which one can create a subspace. Note the basis computed by Sage is “row reduced”.

sage: V = VectorSpace(GF(2),8)
sage: S = V.subspace([V([1,1,0,0,0,0,0,0]),V([1,0,0,0,0,1,1,0])])
sage: S.basis()
[
(1, 0, 0, 0, 0, 1, 1, 0),
(0, 1, 0, 0, 0, 1, 1, 0)
]
sage: S.dimension()
2

Matrix powers

How do I compute matrix powers in Sage? The syntax is illustrated by the example below.

sage: R = IntegerModRing(51)
sage: M = MatrixSpace(R,3,3)
sage: A = M([1,2,3, 4,5,6, 7,8,9])
sage: A^1000*A^1007

[ 3  3  3]
[18  0 33]
[33 48 12]
sage: A^2007

[ 3  3  3]
[18  0 33]
[33 48 12]

Kernels

The kernel is computed by applying the kernel method to the matrix object. The following examples illustrate the syntax.

sage: M = MatrixSpace(IntegerRing(),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]

A kernel of dimension one over \(\QQ\):

sage: A = MatrixSpace(RationalField(),3)(range(9))
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2  1]

A trivial kernel:

sage: A = MatrixSpace(RationalField(),2)([1,2,3,4])
sage: A.kernel()
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
sage: M = MatrixSpace(RationalField(),0,2)(0)
sage: M
[]
sage: M.kernel()
Vector space of degree 0 and dimension 0 over Rational Field
Basis matrix:
[]
sage: M = MatrixSpace(RationalField(),2,0)(0)
sage: M.kernel()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]

Kernel of a zero matrix:

sage: A = MatrixSpace(RationalField(),2)(0)
sage: A.kernel()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]

Kernel of a non-square matrix:

sage: A = MatrixSpace(RationalField(),3,2)(range(6))
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2  1]

The 2-dimensional kernel of a matrix over a cyclotomic field:

sage: K = CyclotomicField(12); a = K.gen()
sage: M = MatrixSpace(K,4,2)([1,-1, 0,-2, 0,-a^2-1, 0,a^2-1])
sage: M
[             1            -1]
[             0            -2]
[             0 -zeta12^2 - 1]
[             0  zeta12^2 - 1]
sage: M.kernel()
Vector space of degree 4 and dimension 2 over Cyclotomic Field of order 12
 and degree 4
Basis matrix:
[               0                1                0     -2*zeta12^2]
[               0                0                1 -2*zeta12^2 + 1]

A nontrivial kernel over a complicated base field.

sage: K = FractionField(PolynomialRing(RationalField(),2,'x'))
sage: M = MatrixSpace(K, 2)([[K.gen(1),K.gen(0)], [K.gen(1), K.gen(0)]])
sage: M
[x1 x0]
[x1 x0]
sage: M.kernel()
Vector space of degree 2 and dimension 1 over Fraction Field of Multivariate
Polynomial Ring in x0, x1 over Rational Field
Basis matrix:
 [ 1 -1]

Other methods for integer matrices are elementary_divisors, smith_form (for the Smith normal form), echelon_form for the Hermite normal form, frobenius for the Frobenius normal form (rational canonical form).

There are many methods for matrices over a field such as \(\QQ\) or a finite field: row_span, nullity, transpose, swap_rows, matrix_from_columns, matrix_from_rows, among many others.

See the file matrix.py for further details.

Eigenvectors and eigenvalues

How do you compute eigenvalues and eigenvectors using Sage?

Sage has a full range of functions for computing eigenvalues and both left and right eigenvectors and eigenspaces. If our matrix is \(A\), then the eigenmatrix_right (resp. eightmatrix_left) command also gives matrices \(D\) and \(P\) such that \(AP=PD\) (resp. \(PA=DP\).)

sage: A = matrix(QQ, [[1,1,0],[0,2,0],[0,0,3]])
sage: A
[1 1 0]
[0 2 0]
[0 0 3]
sage: A.eigenvalues()
[3, 2, 1]
sage: A.eigenvectors_right()
[(3, [
(0, 0, 1)
], 1), (2, [
(1, 1, 0)
], 1), (1, [
(1, 0, 0)
], 1)]

sage: A.eigenspaces_right()
[
(3, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[0 0 1]),
(2, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[1 1 0]),
(1, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[1 0 0])
]

sage: D, P = A.eigenmatrix_right()
sage: D
[3 0 0]
[0 2 0]
[0 0 1]
sage: P
[0 1 1]
[0 1 0]
[1 0 0]
sage: A*P == P*D
True

For eigenvalues outside the fraction field of the base ring of the matrix, you can choose to have all the eigenspaces output when the algebraic closure of the field is implemented, such as the algebraic numbers, QQbar. Or you may request just a single eigenspace for each irreducible factor of the characteristic polynomial, since the others may be formed through Galois conjugation. The eigenvalues of the matrix below are $pmsqrt{3}$ and we exhibit each possible output.

Also, currently Sage does not implement multiprecision numerical eigenvalues and eigenvectors, so calling the eigen functions on a matrix from CC or RR will probably give inaccurate and nonsensical results (a warning is also printed). Eigenvalues and eigenvectors of matrices with floating point entries (over CDF and RDF) can be obtained with the “eigenmatrix” commands.

sage: MS = MatrixSpace(QQ, 2, 2)
sage: A = MS([1,-4,1, -1])
sage: A.eigenspaces_left(format='all')
[
(-1.732050807568878?*I, Vector space of degree 2 and dimension 1 over Algebraic Field
User basis matrix:
[                        1 -1 - 1.732050807568878?*I]),
(1.732050807568878?*I, Vector space of degree 2 and dimension 1 over Algebraic Field
User basis matrix:
[                        1 -1 + 1.732050807568878?*I])
]
sage: A.eigenspaces_left(format='galois')
[
(a0, Vector space of degree 2 and dimension 1 over Number Field in a0 with defining polynomial x^2 + 3
User basis matrix:
[     1 a0 - 1])
]

Another approach is to use the interface with Maxima:

sage: A = maxima("matrix ([1, -4], [1, -1])")
sage: eig = A.eigenvectors()
sage: eig
[[[-sqrt(3)*%i,sqrt(3)*%i],[1,1]],[[[1,(sqrt(3)*%i+1)/4]],[[1,-(sqrt(3)*%i-1)/4]]]]

This tells us that \(\vec{v}_1 = [1,(\sqrt{3}i + 1)/4]\) is an eigenvector of \(\lambda_1 = - \sqrt{3}i\) (which occurs with multiplicity one) and \(\vec{v}_2 = [1,(-\sqrt{3}i + 1)/4]\) is an eigenvector of \(\lambda_2 = \sqrt{3}i\) (which also occurs with multiplicity one).

Here are two more examples:

sage: A = maxima("matrix ([11, 0, 0], [1, 11, 0], [1, 3, 2])")
sage: A.eigenvectors()
[[[2,11],[1,2]],[[[0,0,1]],[[0,1,1/3]]]]
sage: A = maxima("matrix ([-1, 0, 0], [1, -1, 0], [1, 3, 2])")
sage: A.eigenvectors()
[[[-1,2],[2,1]],[[[0,1,-1]],[[0,0,1]]]]

Warning: Notice how the ordering of the output is reversed, though the matrices are almost the same.

Finally, you can use Sage’s GAP interface as well to compute “rational” eigenvalues and eigenvectors:

sage: print(gap.eval("A := [[1,2,3],[4,5,6],[7,8,9]]"))
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
sage: print(gap.eval("v := Eigenvectors( Rationals,A)"))
[ [ 1, -2, 1 ] ]
sage: print(gap.eval("lambda := Eigenvalues( Rationals,A)"))
[ 0 ]

Row reduction

The row reduced echelon form of a matrix is computed as in the following example.

sage: M = MatrixSpace(RationalField(),2,3)
sage: A = M([1,2,3, 4,5,6])
sage: A
[1 2 3]
[4 5 6]
sage: A.parent()
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: A[0,2] = 389
sage: A
[  1   2 389]
[  4   5   6]
sage: A.echelon_form()
[      1       0 -1933/3]
[      0       1  1550/3]

Characteristic polynomial

The characteristic polynomial is a Sage method for square matrices.

First a matrix over \(\ZZ\):

sage: A = MatrixSpace(IntegerRing(),2)( [[1,2], [3,4]] )
sage: f = A.charpoly()
sage: f
x^2 - 5*x - 2
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring

We compute the characteristic polynomial of a matrix over the polynomial ring \(\ZZ[a]\):

sage: R = PolynomialRing(IntegerRing(),'a'); a = R.gen()
sage: M = MatrixSpace(R,2)([[a,1], [a,a+1]])
sage: M
[    a     1]
[    a a + 1]
sage: f = M.charpoly()
sage: f
x^2 + (-2*a - 1)*x + a^2
sage: f.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in a over
Integer Ring

sage: M.trace()
2*a + 1
sage: M.determinant()
a^2

We compute the characteristic polynomial of a matrix over the multi-variate polynomial ring \(\ZZ[u,v]\):

sage: R.<u,v> = PolynomialRing(ZZ,2)
sage: A = MatrixSpace(R,2)([u,v,u^2,v^2])
sage: f = A.charpoly(); f
x^2 + (-v^2 - u)*x - u^2*v + u*v^2

It’s a little difficult to distinguish the variables. To fix this, we might want to rename the indeterminate “Z”, which we can easily do as follows:

sage: f = A.charpoly('Z'); f
Z^2 + (-v^2 - u)*Z - u^2*v + u*v^2

Solving systems of linear equations

Using maxima, you can easily solve linear equations:

sage: var('a,b,c')
(a, b, c)
sage: eqn = [a+b*c==1, b-a*c==0, a+b==5]
sage: s = solve(eqn, a,b,c); s
[[a == -1/4*I*sqrt(79) + 11/4, b == 1/4*I*sqrt(79) + 9/4, c == 1/10*I*sqrt(79) + 1/10], [a == 1/4*I*sqrt(79) + 11/4, b == -1/4*I*sqrt(79) + 9/4, c == -1/10*I*sqrt(79) + 1/10]]

You can even nicely typeset the solution in LaTeX:

sage.: print(latex(s))
...

To have the above appear onscreen via xdvi, type view(s).

You can also solve linear equations symbolically using the solve command:

sage: var('x,y,z,a')
(x, y, z, a)
sage: eqns = [x + z == y, 2*a*x - y == 2*a^2, y - 2*z == 2]
sage: solve(eqns, x, y, z)
[[x == a + 1, y == 2*a, z == a - 1]]

Here is a numerical Numpy example:

sage: from numpy import arange, eye, linalg
sage: A = eye(10)       ##   the 10x10 identity matrix
sage: b = arange(1,11)
sage: x = linalg.solve(A,b)

Another way to solve a system numerically is to use Sage’s octave interface:

sage: M33 = MatrixSpace(QQ,3,3)
sage: A   = M33([1,2,3,4,5,6,7,8,0])
sage: V3  = VectorSpace(QQ,3)
sage: b   = V3([1,2,3])
sage: octave.solve_linear_system(A,b)    # optional - octave
[-0.333333, 0.666667, 0]