Diagonal Matrix is Symmetric

Theorem

Let $D$ be a diagonal matrix.

Then $D$ is symmetric.

Proof

By definition of diagonal matrix:

$\forall j, k: j \ne k \implies a_{jk} = 0 = a_{kj}$

So by definition of transpose of $D$:

$D = D^\intercal$

where $D^\intercal$ denotes the transpose.

Hence the result, by definition of symmetric matrix.

$\blacksquare$