linear algebra - How can I find the dimension of an eigenspace?

I have the following square matrix

$$ A = \begin{bmatrix} 2 & 0 & 0 \\ 6 & -1 & 0 \\ 1 & 3 &-1 \end{bmatrix} $$

I found the eigenvalues:

• $2$ with algebraic and geometric multiplicity $1$ and eigenvector $(1,2,7/3)$.




• $-1$ with algebraic multiplicity $2$ and geometric multiplicity $1$; one eigenvector is $(0,0,1)$.

Thus, matrix $A$ is not diagonizable. My questions are:

1. How can I find the Jordan normal form?




2. How I can find the dimension of the eigenspace of eigenvalue $-1$?




3. In Sagemath, how can I find the dimension of the eigenspace of eigenvalue $-1$?

Answer

The SageMath commands to compute anything about this matrix
are easy to discover.

Define the matrix:

sage: a = matrix(ZZ, 3, [2, 0, 0, 6, -1, 0, 1, 3, -1])

and then type a.jor and then a.eig, where
means hit the TAB key. This will show you the
methods that can be applied to a that start with jor
and with eig.

Then, once you found the method a.jordan_form, read its
documentation by typing a.jordan_form? followed by TAB
or ENTER.

You will find that you can call a.jordan_form() to get
the Jordan form, or a.jordan_form(transformation=True)
to also get the transformation matrix.

sage: j, p = a.jordan_form(transformation=True)
sage: j

[ 2| 0  0]
[--+-----]
[ 0|-1  1]
[ 0| 0 -1]
sage: p
[  1   0   0]
[  2   0   1]
[7/3   3   0]

Here is an exploration of the eigenvalues, eigenspaces,
eigenmatrix, eigenvectors.

sage: a.eigenvalues()
[2, -1, -1]
sage: a.eigenspaces_right()
[
(2, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[  1   2 7/3]),

(-1, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[0 0 1])
]
sage: a.eigenmatrix_right()
(
[ 2  0  0]  [  1   0   0]
[ 0 -1  0]  [  2   0   0]
[ 0  0 -1], [7/3   1   0]
)

sage: j, p
(
[ 2| 0  0]               
[--+-----]  [  1   0   0]
[ 0|-1  1]  [  2   0   1]
[ 0| 0 -1], [7/3   3   0]
)
sage: a.eigenvectors_right()
[(2, [
  (1, 2, 7/3)

  ], 1), (-1, [
  (0, 0, 1)
  ], 2)]