Linear Algebra
SAGE has extensive linear algebra capabilities.
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 arithmetic
Computing matrix powers in Sage 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]
Matrix addition and multiplication:
sage: R = IntegerModRing(51)
sage: M = MatrixSpace(R,3,3)
sage: A = M([1,2,3, 4,5,6, 7,8,9])
sage: B = M([1,0,1, 0,1,0, 1,1,1])
sage: A+B
[ 2 2 4]
[ 4 6 6]
[ 8 9 10]
sage: A*B
[ 4 5 4]
[10 11 10]
[16 17 16]
sage: -7*B
[44 0 44]
[ 0 44 0]
[44 44 44]
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
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 dimension 2 over Rational Field
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(MPolynomialRing(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 include: elementary_divisors, smith_form (for the Smith normal form), echelon (a method for integer matrices) for the Hermite normal form, frobenius for the Frobenius normal form (rational canonical form).
There a many methods for matrices over a field such as
Eigenvectors and eigenvalues
How do you compute eigenvalues and eigenvectors using Sage?
Sage included both in the eigenspaces command, the output of which has several components, corresponding to the different eigenvalues.
sage: MS = MatrixSpace(QQ, 3, 3)
sage: A = MS([[1,1,0],[0,2,0],[0,0, 3]])
sage: A
[1 1 0]
[0 2 0]
[0 0 3]
sage: A.eigenspaces()
[
(3, [
(0, 0, 1)
]),
(2, [
(0, 1, 0)
]),
(1, [
(1, -1, 0)
])
]
sage: v1*A == r1*v1
True
sage: v2*A == r2*v2
True
sage: v3*A == r3*v3
True
Note that these are left eigenspaces. To get the right eigenspaces, first take the transpose:
sage: At = A.transpose()
sage: v1 = At.eigenspaces()[0][1][0]
sage: v2 = At.eigenspaces()[1][1][0]
sage: v3 = At.eigenspaces()[2][1][0]
sage: A*v1 == r1*v1
True
sage: A*v2 == r2*v2
True
sage: A*v3 == r3*v3
True
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
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
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
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 + (-1*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". However, as of Sage 1.5, we can no longer reassign the name of a determinate in an instantiated polynomial ring. However, we can locally change the name in a with block:
sage: with localvars(f.parent(), 'Z'):
....: print(f)
Z^2 + (-1*v^2 - u)*Z - u^2*v + u*v^2