Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $(1+\sqrt{2}i),(1-\sqrt{2}i)$

share|improve this question
1  
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
add comment

2 Answers

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.

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.