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sanctifier
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About the invariance of similar linear operators and their minimal polynomial
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 mτ(x) = mσ(x), i.e., the minimal polynomial is an invariant under similarity of operators.
I wonder how it's proved.
Thanks for any help!
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 mτ(x) = mσ(x), i.e., the minimal polynomial is an invariant under similarity of operators.
I wonder how it's proved.
Thanks for any help!