Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: A = matrix(QQ, [[1, 0, 1, 1, -1],
... [0, 1, 0, 4, 8],
... [2, 1, 3, 5, 1],
... [2, -1, 1, 0, -2],
... [0, -1, -1, -5, -8]])
sage: [e in QQ for e in A.eigenvalues()]
[False, False, False, False, False]
sage: A.is_diagonalizable()
Traceback (most recent call last):
...
RuntimeError: an eigenvalue of the matrix is not contained in Rational Field
sage: [e in QQbar for e in A.eigenvalues()]
[True, True, True, True, True]
sage: A.is_diagonalizable(base_field=QQbar)
True
Other exact fields may be employed, though it will not always be possible to expand their base fields to
contain all the eigenvalues.
sage: F. = FiniteField(5^2)
sage: A = matrix(F, [[ 4, 3*b + 2, 3*b + 1, 3*b + 4],
... [2*b + 1, 4*b, 0, 2],
... [ 4*b, b + 2, 2*b + 3, 3],
... [ 2*b, 3*b, 4*b + 4, 3*b + 3]])
sage: A.jordan_form()
[ 4 1| 0 0]
[ 0 4| 0 0]
[---------------+---------------]
[ 0 0|2*b + 1 1]
[ 0 0| 0 2*b + 1]
sage: A.is_diagonalizable()
False
sage: F. = QuadraticField(-7)
sage: A = matrix(F, [[ c + 3, 2*c - 2, -2*c + 2, c - 1],
... [2*c + 10, 13*c + 15, -13*c - 17, 11*c + 31],
... [2*c + 10, 14*c + 10, -14*c - 12, 12*c + 30],
... [ 0, 2*c - 2, -2*c + 2, 2*c + 2]])
sage: A.jordan_form(subdivide=False)
[ 4 0 0 0]
[ 0 -2 0 0]
[ 0 0 c + 3 0]
[ 0 0 0 c + 3]
sage: A.is_diagonalizable()
True
A trivial matrix is diagonalizable, trivially.
sage: A = matrix(QQ, 0, 0)
sage: A.is_diagonalizable()
True
A matrix must be square to be diagonalizable.
sage: A = matrix(QQ, 3, 4)
sage: A.is_diagonalizable()
False
186 Chapter 7. Base class for matrices, part 2