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Find all $3\times 3$ matrices that commute with

$$A =\left( \begin{array}{cc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3\end{array} \right)$$

My progress:

I know that a I need to find a matrix such that $AX = XA$. However I'm getting stuck when:

$$AX =\left( \begin{array}{cc} a_1x_{11} & a_1x_{12} & a_1x_{13}\\ a_2x_{21} & a_2x_{22} & a_2x_{23}\\ a_3x_{31} & a_3x_{32} & a_2x_{33}\end{array} \right)$$

$$XA =\left( \begin{array}{cc} a_1x_{11} & a_2x_{12} & a_3x_{13}\\ a_1x_{21} & a_2x_{22} & a_3x_{23}\\ a_1x_{31} & a_2x_{32} & a_3x_{33}\end{array} \right)$$

The answer has been given as:

$$\left( \begin{array}{cc} b_1 & 0 & 0 \\ 0 & b_2 & 0 \\ 0 & 0 & b_3 \end{array} \right)$$

I don't understand how they're getting that form. Can someone please explain?

Patryk
  • 123

1 Answers1

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Assuming that all the $a_i$ are different, it's really easy. Just look at each entry in $XA$ and the corresponding entry in $AX$, and you'll see the answer. For instance, we need $a_1x_{12}=a_2x_{12}$, which can only happen if $x_{12}=0$.

Arthur
  • 199,419