Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: a=matrix([(1, 9, -1, -1), (-2, 0, -10, 2), (-1, 0, 15, -2), (0, 1, 0, -1)])
sage: a.eigenvalues()
[-0.9386318578049146, 15.50655435353258, 0.2160387521361705 - 4.713151979747493*I, 0.216
A symmetric matrix a+a.transpose() should have real eigenvalues
sage: b=a+a.transpose()
sage: ev = b.eigenvalues(); ev
[-8.35066086057957, -1.107247901349379, 5.718651326708515, 33.73925743522043]
The eigenvalues are elements of QQbar, so they really represent exact roots of polynomials, not just approximations.
sage: e = ev[0]; e
-8.35066086057957
sage: p = e.minpoly(); p
x^4 - 30*x^3 - 171*x^2 + 1460*x + 1784
sage: p(e) == 0
True
To perform computations on the eigenvalue as an element of a number field, you can always convert back
to a number field element.
sage: e.as_number_field_element()
(Number Field in a with defining polynomial y^4 - 2*y^3 - 507*y^2 - 3972*y - 4264,
a + 7,
Ring morphism:
From: Number Field in a with defining polynomial y^4 - 2*y^3 - 507*y^2 - 3972*y - 4264
To: Algebraic Real Field
Defn: a |--> -15.35066086057957)
Notice the effect of the extend option.
sage: M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]])
sage: M.eigenvalues()
[2, -1*I, 1*I]
sage: M.eigenvalues(extend=False)
[2]
eigenvectors_left(extend=True)
Compute the left eigenvectors of a matrix.
For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of
eigenvectors forming a basis for the corresponding left eigenspace, and n is the algebraic multiplicity of
the eigenvalue.
If the option extend is set to False, then only the eigenvalues that live in the base ring are considered.
EXAMPLES: We compute the left eigenvectors of a 3 × 3 rational matrix.
sage: A = matrix(QQ,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: es = A.eigenvectors_left(); es
[(0, [
(1, -2, 1)
], 1),
(-1.348469228349535, [(1, 0.3101020514433644, -0.3797958971132713)], 1),
(13.34846922834954, [(1, 1.289897948556636, 1.579795897113272)], 1)]
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