I was thinking that If a matrix is diagonal with entries belong to $ \mathbb{R}$ then it has to be a symmetric i.e. $ A^T = A$, but can I say that every diagonal matrix is orthogonal? I'm confused about this.
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$\begingroup$Not all real diagonal matrices are orthogonal. As you said, they are indeed self-adjoint ($A^t=A$), but that's not the same as orthogonality - $AA^t=I$.
If I want a diagonal matrix to be orthogonal, I need its inverse to be itself (since we have seen that diagonal matrices are self adjoint). Meaning - $A=A^{-1}$. There are only two matrices that satisfy this: $I$ and $-I$.
$\endgroup$ $\begingroup$Hint
Consider the definition of orthogonal matrix and apply it on $D$, considering directly its diagonal entries.
Partial Answer
Not all the diagonal matrices are orthogonal.
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