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Math Help - Invariant Subspace Problem

  1. #1
    Member
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    Mar 2008
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    Acolman, Mexico
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    Invariant Subspace Problem

    Hello, I need help with this problem

    Let A be a n \times n matrix, B a k \times k and X a n \times k such that AX=XB.
    Show:

    1. X is an invariant subspace relative to A.
    2. If X is of column rank k, the every eigenvalue of B is an eigenvalue of A.


    Thanks in advance.
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  2. #2
    Member
    Joined
    Mar 2009
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    Invariant Subspace Problem

    1. X is an invariant subspace relative to A.
    I suppose by this wording you mean that the subspace \mathop{\textrm{Ran}}X is invariant under A?

    Let u\in\mathop{\textrm{Ran}}X. Then u=Xc for some c\in\mathbb R^k.

    Hence Au=AXc=XBc and so Au\in\mathop{\textrm{Ran}}X; \mathop{\textrm{Ran}}X is therefore invariant under A.

    2. If X is of column rank k, the every eigenvalue of B is an eigenvalue of A.
    The condition on the column rank implies that if v\in\mathbb R^k is non-zero then Xv is also non-zero.

    Now let \lambda be an eigenvalue of B with non-zero eigenvector v\in\mathbb R^k.

    Let u=Xv so that, by the remark above, u is non-zero. Then

    Au=AXv=XBv=X(\lambda v)=\lambda Xv=\lambda u.

    Since u is non-zero, clearly \lambda is an eigenvalue of A.
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