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Let $A$ be the matrix $$A = \begin{pmatrix} 1 & \sqrt{2}\\ -\sqrt{2} & -1\\ \end{pmatrix}$$

Compute the matrix $B = 3A -2A^2 - A^3 -5A^4 + A^6$.

Could any one give me any hint for this one? I have calculated the eigenvalues they are $(1+\sqrt{2}i),(1-\sqrt{2}i)$

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hint: compute directly $A^2$. – Raymond Manzoni Dec 18 '12 at 14:22
Find a matrix $P$ to diagonalize the matrix. You can then easily add up the diagonialized matrices and then use $P^{-1}$. Or, you can just brute force it! – Zach L. Dec 18 '12 at 14:29
up vote 1 down vote accepted

We have the characteristic polynomial $ch_A(x)=\begin{vmatrix}1-x&\sqrt{2}\\-\sqrt{2}&1-x\end{vmatrix}=(x-1)^2+2=x^2-2x+3$.

Then by Cayley-Hamilton theorem we know that $A^2-2A+3=0$.

So if you divide your polynomial $p(x)$ by $(x^2-2x+3)$ and you get $p(x)=q(x)(x^2-2x+3)+r(x)$, you only need to calculate $r(A)$, i.e., to plug the matrix $A$ into the remainder.

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In this example probably the best way is to use Raymond Manzoni's hint.

In general if $A$ is similar to $C$ with $A=P^{-1}CP$ for some invertible $P$ then for any polynomial $\phi(x)$: $$\phi(A)=P^{-1}\phi(C)P.$$ In your case use that $A$ is similar to $\begin{pmatrix}1+\sqrt{2}&0\\0&1-\sqrt{2}\end{pmatrix}$.
Also see this.

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