
cayleyHamilton law
1)
there is $\displaystyle A=\left(\begin{array}{ccc}4 & 2 & 4\\2 & 1 & 2\\4 & 2 & 4\end{array}\right)$
A.we define $\displaystyle f:R^{3}xR^{3}>R$ by $\displaystyle f(x,y)=(Ax,y)$ for every $\displaystyle y=(y_{1},y_{2},y_{3})^{t}$and $\displaystyle x=(x_{1},x_{2},x_{3})^{t}$ in $\displaystyle R^{3}$
does f define inner product in $\displaystyle R^{3}$?
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)
$\displaystyle (v,v)\geq0$
i dont know how to apply these rules here
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B.find a simetric matrices B so $\displaystyle B^{2}=A $
(clue: use cayleyHamilton law)
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how i tried:
hamilton law say that the representative matrice turns to zero the minimal polinomial M(A)=0
any guidence
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Re: cayleyHamilton law
a) are you sure about the definition of f? an inner product should go from $\displaystyle \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ .
perhaps you mean $\displaystyle f(x,y) = <(Ax)^T,y> $? without some further clarification, your question doesn't make much sense.

Re: cayleyHamilton law
yes you are correct i had typing mistake its
$\displaystyle f:R^{3}xR^{3}>R$