Getting a simultaneous basis of diagonalization for a list of diagonalizable matrices that
pairwise commute is just a matter of getting a diagonalization basis for any one of them!
This can be obtained with jordan_form
with the argument transformation=True
.
For example, starting from these two matrices a
and b
:
sage: a = matrix(QQ, [[1, 1, 1], [-2, 3, 2], [0, 1, 2]])
sage: b = matrix(QQ, [[-1, 1, 1], [1, 1, -1], [-3, 1, 3]])
Check that they commute:
sage: a * b
[-3 3 3]
[-1 3 1]
[-5 3 5]
sage: b * a
[-3 3 3]
[-1 3 1]
[-5 3 5]
sage: a * b == b * a
True
If we diagonalize them separately:
sage: da, pa = a.jordan_form(transformation=True)
sage: db, pb = b.jordan_form(transformation=True)
we can notice that the transformation matrices pa
and pb
have the same columns, permuted:
sage: pa
[ 1 1 0]
[ 1 0 1]
[ 1 1 -1]
sage: pb
[ 0 1 1]
[ 1 1 0]
[-1 1 1]
Obtaining the simultaneous diagonal form is just a matter of reordering the columns of:
sage: da
[3|0|0]
[-+-+-]
[0|2|0]
[-+-+-]
[0|0|1]
sage: db
[2|0|0]
[-+-+-]
[0|1|0]
[-+-+-]
[0|0|0]
For instance, use the transformation matrix pa
to get simultaneous diagonalizations:
sage: qa = ~pa
sage: qa * a * pa
[3 0 0]
[0 2 0]
[0 0 1]
sage: qa * b * pa
[1 0 0]
[0 0 0]
[0 0 2]
could you write the code?
Please give us at least two of the many commuting matrices that can be diagonalized (simultaneously).
Let me join the club of asksage junkies : we need your code to understand your problem and (hopefully) provide a solution.
Please provide an example of matrices for the computation you are asking about.
Ready-to-copy-paste examples make it way easier to explore a question, thereby increasing chances of an answer.