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 \mathbb{Q} :

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 \mathbb{Q} or a finite field: row_span, nullity, transpose, swap_rows, matrix_from_columns, matrix_from_rows, among many others.

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