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**Astrid** Find the minimal polynomial given $\displaystyle T:\mathbb R^2\to\mathbb R^2$ where $\displaystyle T(x,y)=(x+y,x-y).$

I found the associated matrix, which is $\displaystyle A=\left[\begin{array}{rr}1&1\\1&-1\end{array}\right].$

Now I need to compute the characteristic polynomial, and this is $\displaystyle -\left( 1-\lambda ^{2} \right)-1=\lambda ^{2}-2=\big(\lambda-\sqrt2\big)\big(\lambda+\sqrt2\big).$

Now suppose I take $\displaystyle P(\lambda)=\lambda-\sqrt2$ then I make $\displaystyle P(A),$ if this is zero, then that polynomial is the minimal one right? If is not zero, then I consider the characteristic polynomial, and that's actually the minimal polynomial because of simple application of Cayley - Hamilton Theorem.

== Yup, but in fact it's easier: since the minimal and the char. polynomials have the same irreducible factors, you know the min. pol. = the char. pol. in this case!

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Another question I have, suppose we're computing the characteristic polynomial, when is it equal to the minimal one?