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I am in a situation where I need to prove a property, and I need to know this:

Is the product of a diagonal matrix and an orthogonal matrix commutative?

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closed as off-topic by amWhy, GNUSupporter 8964民主女神 地下教會, José Carlos Santos, Davide Giraudo, Cesareo Dec 22 '18 at 0:12

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    $\begingroup$ The simplest example like $\begin{bmatrix}1&0\\0&0\end{bmatrix}$ and $\begin{bmatrix}0&1\\1&0\end{bmatrix}$ will tell you this is false. $\endgroup$ – Christoph Dec 9 '18 at 14:05
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take $$A = \begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\\\end{bmatrix}$$

and $$B = \begin{bmatrix}2&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&2\end{bmatrix}$$

Then $A$ is orthogonal matrix and $B$ is diagonal But, $AB≠BA$.

(You can see, here $A$ is also permutation matrix corresponding to permutation $\sigma =(1324)$. Hence accordingly pre multiplying by $A$ to $B$ just permuted rows of $B$ and post multiplying $A$ to $B$ just permuted columns of $B$)

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