Search the corpus

    The minimal polynomial of a restriction divides the minimal polynomial

    E-HK-64-2

    Area
    Algebra
    Topics
    Annihilating Polynomials, Invariant Subspaces, Linear Algebra
    Status
    draft

    Let \(W\) be an invariant subspace for \(T\). Prove that the minimal polynomial for the restriction operator \(T_W\) divides the minimal polynomial for \(T\), without referring to matrices.

    Solution

    Let \(m_T\) be the minimal polynomial of \(T\). Since \(m_T(T)=0\), for every \(w\in W\) we have \[ m_T(T_W)w=m_T(T)w=0, \] because \(W\) is invariant and therefore every power of \(T_W\) is the restriction to \(W\) of the corresponding power of \(T\). Hence \(m_T\) annihilates \(T_W\). By the defining divisibility property of the minimal polynomial, \[ m_{T_W}\mid m_T. \] No matrices are involved.

    Source Collections