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Math Help - Cayley-Hamilton Theorem for matrices

  1. #1
    Member Last_Singularity's Avatar
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    Cayley-Hamilton Theorem for matrices

    The Cayley Hamilton Theorem states that if T is a linear operation on vector space V and f(t) is the characteristic polynomial of T, then f(T) is the zero transformation.

    How do I extend this to matrices? As in, how do I show that: if A is n \times n and f(t) is the characteristic of A, then f(A) is the zero matrix?

    I attempted to prove it using f(A) = det(A - AI) = det(O) = 0 but this reasoning seems faulty...
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  2. #2
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    Remember that linear transformations between finite dimensional vector spaces are in 1-1 correspondence with matrices (and they do the same thing with the exception that the matrices act on \mathbb{F} ^n ) so translate everything from matrices to lin.tranf. and see what f(A) does in \mathbb{F} ^n
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