About the invariance of similar linear operators and their minimal polynomial(adsbygoogle = window.adsbygoogle || []).push({});

Notations:

F denotes a field

V denotes a vector space over F

L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions.

τ, σ, φ denote distinct linear operators contained in L(V)

m_{?}(x) denotes the minimal polynomial of the linear operator "?"

～ denotes the similarity of the left and right operand, e.g., if τ ～ σ, then τ = φσφ^{-1}

Question:

If τ ～ σ are similar linear operators on V, then m_{τ}(x) = m_{σ}(x), i.e., the minimal polynomial is an invariant under similarity of operators.

I wonder how it's proved.

Thanks for any help!

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# About the invariance of similar linear operators and their minimal polynomial

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