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What are values of $a,b,c$ such that matrix $A$ and $B$ commutes, i.e. $AB=BA$, where $A$ and $B$ are (3,3) matrix such that $A=\pmatrix{a_0 & b_0 &c_0\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2}$

$$B=\pmatrix{1 & 1 &1\\ 1 & \omega & \omega^2\\ 1 & \omega^2 & \omega}$$ where $\omega+\omega^2+1=0$

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I've put your equations into LaTeX; hopefully I haven't altered your intended meaning. Also, if this is homework, please add the "homework" tag to your question. –  Zev Chonoles Jun 19 '11 at 4:37
    
If I understand your question correctly, isn't $$A=\pmatrix{0 & 0 &0\\ 0 & 0 &0\\ 0& 0 & 0}$$ a valid answer, because $$AB=\pmatrix{0 & 0 &0\\ 0 & 0 &0\\ 0& 0 & 0}=BA?$$ Or is the question asking you to find every matrix $A$ for which $A$ and $B$ commute? –  Zev Chonoles Jun 19 '11 at 5:16
    
Thanks for LaTeX. Yes, identity or zero matrix satisfies the condition but I want to find general form. –  user12290 Jun 19 '11 at 5:37
    
You can define a 9-component vector v composed of those a, b, c coefficients, then write AB-BA=0 as a linear system Xv=0, and then v is any vector in the kernel space of X. Also, you might need to specify that $ \omega $ is one of $ \exp(2\pi i / 3) $ or $ \exp(4\pi i /3) $ and not the other, as they are both solutions to the quadratic. –  anon Jun 19 '11 at 5:42
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2 Answers 2

up vote 8 down vote accepted

Yes, as the other answer notes, B has distinct eigenvalues. Therefore, anything commuting with it must be a polynomial in B. (Degrees no higher than 2 suffice.)

In some detail: let T be a matrix so that $TBT^{-1}$ is diagonal, with distinct diagonal entries $b_1, b_2, b_3$. Since $AB=BA$, $(TAT^{-1})(TBT^{-1})=(TBT^{-1})(TAT^{-1})$. For any matrix $M$, the $ij$ entry of $M(TBT^{-1})$ is $m_{ij}b_j$, while the $ij$ entry of $(TBT^{-1})M$ is $b_im_{ij}$. Thus, a matrix commuting with $TBT^{-1}$ is diagonal. Use Lagrange interpolation to find a quadratic polynomial $P$ so that $P(TBT^{-1})=TAT^{-1}$ is that other polynomial. Happily, $P(TBT^{-1})=T.P(B).T^{-1}$, so $A=P(B)$.

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It is clear that polynomial in B must be commutative with B. Will you kindly explain why converse is true? –  user12290 Jun 20 '11 at 14:47
    
@user12290: Let the eigenvalues of $B$ be $\lambda_i, i=1,2,3$. Anything that commutes with $B$ must map an eigenspace of $B$ to itself: if $Bx=\lambda_i x$, then $BAx=ABx=\lambda_i Ax$. Here the eigenspaces are 1-dimensional, so $A$ must act on every one of them as a scalar. IOW $A$ is determined by the three scalars $\mu_i,i=1,2,3$. By Lagrange's interpolation formula there exists a quadratic polynomial $p(x)$ such that $p(\lambda_i)=\mu_i$ for all $i=1,2,3$. Thus $A=p(B)$. –  Jyrki Lahtonen Jun 21 '11 at 13:05
    
Thanks for the details. –  user12290 Jun 24 '11 at 5:16
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It turns out that $B$ has three different eigenvalues, so in a suitable basis it assumes diagonal form $D$. If $\hat A$ is the correspondingly transformed matrix $A$ then $AB=BA$ iff $\hat A D=D\hat A$. Now you have a simpler problem.

As you can see in Paul Garrett's answer you don't even have to compute the eigenvectors of $B$ and get a parametric representation of all $A$'s commuting with $B$ for free.

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Will you kindly give the details? –  user12290 Jun 20 '11 at 11:49
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