<s>[INST] Find the minimal polynomial of $(x-1)^2(x-2)^3(x-3)$? Given ch. polynomial is $(x-1)^2(x-2)^3(x-3)$ and for the eigenvalue $1$, we have one eigenvector, for eigenvalue $2$ we have $2$ eigenvector and for $3$ we have $1$ eigenvector. Now determine the minimal polynomial?
I know that minimal polynomial divides the ch. polynomial, so I list the probable candidates as $$(x-1)(x-2)(x-3)$$ $$(x-1)(x-2)^2(x-3)$$ $$(x-1)^2(x-2)^2(x-3)$$ and $$(x-1)^2(x-2)(x-3).$$ But how to determine which one of these serves the purpose? There is no additional information given. Help me out
 [/INST] In fact there exist operators having your prescribed characteristic polynomial for each of the possible minimal polynomials provided:
Indeed, 


*

*for $(x−1)(x−2)(x−3)$ we have the diagonal matrix diag$(1,1,2,2,2,3)$

*for $(x−1)^2(x−2)^2(x−3)$; 
$$\left(\begin{smallmatrix}1 & 1& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 0 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
which is unique up to permutation of the Jordan blocks.

*for $(x−1)(x−2)^2(x−3)$; 
$$\left(\begin{smallmatrix}1 & 0& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 0 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
again unique up to permutation of the Jordan blocks.
and finally


*

*for $(x−1)^2(x−2)^3(x−3)$; 
$$\left(\begin{smallmatrix}1 & 1& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 1 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
up to permutation of the Jordan blocks.


Additionally, the combinatorial theory behind these ideas is very interesting. What you have specified in your problem (which is merely a characteristic polynomial) is 


*

*the dimension of the vector space (a.k.a. the size of the matrix in question); this is the degree of the characteristic polynomial.

*the algebraic multiplicity of each of the eigenvalues occurring in its upper triangular form (i.e. exactly which values will appear on the diagonal of the Jordan form)

*The different possible minimal polynomials hold the geometric multiplicity of each of the eigenvalues which specify the size of the maximum Jordan block for each of the eigenvalues in question.
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