Can anybody prove this.
Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial $( − 1)^2( − 2)$
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2Welcome to MSE. What have you tried? What are your issue to tackle the problem? Without showing your own efforts, your question may be closed. – mathcounterexamples.net Jun 03 '22 at 19:45
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Here's where to start: what are the eigenvalues? What their multiplicities? What kinds of Jordan normal forms will satisfy this polynomial, but not, say, $(x - 1)(x - 2)$? – Theo Bendit Jun 03 '22 at 22:24
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Yes somebody can prove this.
The uniqueness does not depend on the particular polynomial, just on the fact that is it monic and of degree equal to the size of the matrix. Then (using the Cayley-Hamilton theorem) the minimal polynomial is also the characteristic polynomial, and this answer tells you that any such matrix is similar to the companion matrix of the polynomial, and hence two such matrices are similar to each other.
And even without invoking Cayley-Hamilton, it can be shown that there always exist vectors $v$ such that the $T$-annihilator of $v$ is actually the minimal polynomial of $T$ (where $T$ is any endomorphism of a finite dimensional vector space, here multiplication by you matrix, and the $T$-annihilator of $v$ is the minimal polynomial $P$ such that $P[T](v)=0$). Once you've got such vectors for both matrices, use bases $[T^i(v)]_{0\leq i<n}$ to perform a change of basis, after which by construction you get a companion matrix for you minimal polynomial in both cases, again showing the matrices are similar (and similar to that companion matrix).
Marc van Leeuwen
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