Why do diagonal matrices commute under multiplication?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In general, matrix multiplication is not commutative, meaning that for two arbitrary matrices $A$ and $B$ , $A B$ does not necessarily equal $B A$ . However, diagonal matrices—square matrices where all elements $a_{i j}$ are zero except when $i = j$ —are a notable exception to this rule.1

Diagonal matrices commute because their multiplication effectively reduces to the independent multiplication of their corresponding diagonal elements. If $A$ and $B$ are diagonal matrices, the element in the $i$ th row and $j$ th column of their product $A B$ is the product of the $i$ th diagonal element of $A$ and the $j$ th diagonal element of $B$ only when $i = j$ ; otherwise, the product is zero. Because the multiplication of these individual diagonal elements (which are ordinary numbers) is commutative, the resulting matrices $A B$ and $B A$ are identical.1