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Math Help - Direct Sums

  1. #1
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    Direct Sums

    "Suppose that T : V -> V is a linear transformation of vector spaces over
    R whose minimal polynomial has no multiple roots. Show that V can be
    expressed as a direct sum

    V = V1 + V2 + + Vt

    of T-stable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T can be represented by an n n matrix with at most 2n non-zero entries, where n := dim(V)."

    If only there was a way to represent the complex roots of the minimal polynomial with 2x2 matrices all call the whole 2x2 matrix an "eigenvalue". If not, I don't know what to do.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Direct Sums

    Think about this.

    If the minimal polynomial is m(x)=\prod_j (x-z_j), we can define V_j^1=\{v: Tv=Re(z_j) v\}, V_j^2=\{v: Tv=Im(z_j) v\}, and V_j=V_j^1+V_j^2.
    Thanks from HallsofIvy
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