Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 1 1 1]
[ -2 0.1303061543300932 3.069693845669907]
[ 1 -0.7393876913398137 5.139387691339814]
sage: A*P == P*D
True
Because P is invertible, A is diagonalizable.
sage: A == P*D*(~P)
True
The matrix P may contain zero columns corresponding to eigenvalues for which the algebraic multiplicity
is greater than the geometric multiplicity. In these cases, the matrix is not diagonalizable.
sage: A = jordan_block(2,3); A
[2 1 0]
[0 2 1]
[0 0 2]
sage: A = jordan_block(2,3)
sage: D, P = A.eigenmatrix_right()
sage: D
[2 0 0]
[0 2 0]
[0 0 2]
sage: P
[1 0 0]
[0 0 0]
[0 0 0]
sage: A*P == P*D
True
TESTS:
For matrices with floating point entries, some platforms will return eigenvectors that are negatives of
those returned by the majority of platforms. This test accounts for that possibility. Running this test
independently, without adjusting the eigenvectors could indicate this situation on your hardware.
sage: B = matrix(QQ, 3, 3, range(9))
sage: em = B.change_ring(RDF).eigenmatrix_right()
sage: evalues = em[0]; evalues.dense_matrix().zero_at(2e-15)
[ 13.3484692283 0.0 0.0]
[ 0.0 -1.34846922835 0.0]
[ 0.0 0.0 0.0]
sage: evectors = em[1];
sage: for i in range(3):
... scale = evectors[0,i].sign()
... evectors.rescale_col(i, scale)
sage: evectors
[ 0.164763817... 0.799699663... 0.408248290...]
[ 0.505774475... 0.104205787... -0.816496580...]
[ 0.846785134... -0.591288087... 0.408248290...]
eigenspaces_left(format=’all’, var=’a’, algebraic_multiplicity=False)
Compute the left eigenspaces of a matrix.
Note that eigenspaces_left() and left_eigenspaces() are identical methods. Here “left”
refers to the eigenvectors being placed to the left of the matrix.
INPUT:
157