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Math Help - cayley-Hamilton law

  1. #1
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    cayley-Hamilton law

    1)
    there is A=\left(\begin{array}{ccc}4 & -2 & 4\\-2 & 1 & -2\\4 & -2 & 4\end{array}\right)
    A.we define f:R^{3}xR^{3}->R by f(x,y)=(Ax,y) for every y=(y_{1},y_{2},y_{3})^{t}and x=(x_{1},x_{2},x_{3})^{t} in R^{3}
    does f define inner product in R^{3}?
    how i tried:
    in the book by definition inner product is following these rules:
    (u,v)=(v,u)
    (u+v,w)=(u,w)+(v,w)
    (ku,v)=k(u,v)
    (v,v)\geq0
    i dont know how to apply these rules here
    ?
    B.find a simetric matrices B so B^{2}=A
    (clue: use cayley-Hamilton law)
    ?
    how i tried:
    hamilton law say that the representative matrice turns to zero the minimal polinomial M(A)=0
    any guidence
    ?
    Last edited by transgalactic; October 1st 2011 at 06:29 AM.
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  2. #2
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    Re: cayley-Hamilton law

    a) are you sure about the definition of f? an inner product should go from \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} .

    perhaps you mean  f(x,y) = <(Ax)^T,y> ? without some further clarification, your question doesn't make much sense.
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  3. #3
    MHF Contributor
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    Re: cayley-Hamilton law

    yes you are correct i had typing mistake its
    f:R^{3}xR^{3}->R
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