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Math Help - Prove that eigenspace is a T-invariant subspace

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    Unhappy Prove that eigenspace is a T-invariant subspace

    Let T: V -> V, V is a finite-dimensional vector space
    T^2 is identity operator.
    lamda be a scalar.
    The eigenspace V^(lamda) is the set of eigen-vectors of T with eigen-value lamda, together with zero.

    Prove that V^(lamda) is a T-invariant subspace.

    Prove that for all v in V, v - Tv is either an eigen-vector with eigen-value -1 or zero vector

    Prove that V is direct sum of the eigenspaces V^(1) and V^(-1)

    Please help
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    Quote Originally Posted by mathbeginner View Post
    Let T: V -> V, V is a finite-dimensional vector space
    T^2 is identity operator.
    lamda be a scalar.
    The eigenspace V^(lamda) is the set of eigen-vectors of T with eigen-value lamda, together with zero.

    Prove that V^(lamda) is a T-invariant subspace.

    Prove that for all v in V, v - Tv is either an eigen-vector with eigen-value -1 or zero vector

    Prove that V is direct sum of the eigenspaces V^(1) and V^(-1


    Please help
    Let E_\lambda be the eigenspace with eigenvalue lambda. Then if
    v \in E_\lambda then

    T(v)=\lambda v; T^2(v)=T(T(v))=T(\lambda v)=\lambda^2v

    Notice that both T(v) \in E_\lambda and T^2(v) \in E_{\lambda}

    w is an eigenvector if T(w)=\lambda w and w \ne 0

    Calculate

    T(v-T(v))=T(v)-T^2(v)=...

    This should get you started.
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