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Cell["\<\
In this worksheet, we explore the primary, SN and cyclic decompositions of \
the following matrix.\
\>", "Text",
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First I compute the minimal polynomial of A by computing p_(T,v) for a \
'suitably random' vector v.  It turns out that the following vector v = v0 is \
random enough:\
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Cell["\<\
It' s not clear at this point whether v0, v1, v2, v3 are independent vectors, \
so I check by making them the columns of a matrix and evaluating the \
nullspace.\
\>", "Text",
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Cell["\<\
That is, the nullspace is trivial, so the vectors are independent.  I proceed \
one more step and check again.\
\>", "Text",
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Cell["\<\
So the vectors are now dependent, and in fact v4 - 2v3 + v2 = 0 is a \
dependence relation.  I infer that p_T,v(x) = x^4 - 2x^3 + x^2.  To see if \
this is the minimal polynomial for A, I evaluate p_T,v(A):\
\>", "Text",
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Cell["\<\
So p_min(x) = x^4 -2 x^3 + x^2 = x^2(x-1)^2, and by the primary decomposition \
theorem, I know that R^4 is a direct sum of the nullspaces of A^2 and \
(A-I)^2.  So I find bases for these and combine them to get a basis B for \
R^8.\
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The matrix for the transformation T (x) = A x  relative to the basis B is \
therefore given by\
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And as expected, the matrix has two 4 x 4 blocks corresponding to the two \
four dimensional invariant subspaces in the primary decomposition.\
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