n-th power of matrices

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You can find an interactive Sage application here: http://shrek.unideb.hu/~tengely/Magya... the last part is "Mátrixok n-edik hatványának zárt alakja", it is in Hungarian, but the mathematics and Sage command will help I think. In case of your example this code will not work, the eigenvalue 1 has multiplicity larger than 1. To make it work the computation related to the eigenvectors should be modified. At the above page the matrix is given by matrix([[1,2,1],[6,-1,0],[-1,-2,-1]]) and after pushing the "Számolás" button you see the general form of the $n$th power of this matrix. If you have a diagonalizable matrix the above code should do the job.

This was asked again in Ask Sage question 35658, and an initial implementation was proposed as an answer to that question, and included in SageMath at Sage Trac ticket 22523.

A bug was reported in Ask Sage question 41622 and a fix for this bug was provided as an answer to that question and is ready for inclusion in SageMath at Sage Trac ticket 25082.

This makes it possible to compute the matrix power in the question here.

sage: n = SR.var('n')
sage: y = matrix([[1, 0, 0], [0, 1, 1], [0, 0, 1]]); y
[1 0 0]
[0 1 1]
[0 0 1]
sage: z = y^n; z
[1 0 0]
[0 1 n]
[0 0 1]

Then one can substitute for particular values:

sage: z.subs({n: 5})
[1 0 0]
[0 1 5]
[0 0 1]
sage: y^5
[1 0 0]
[0 1 5]
[0 0 1]

Unless i misunderstand you question, if M is your matrix and n is an integer, you can just do:

sage: M^n

or, if you prefer to be Python compatible:

sage: M**n