Symmetric matrix - orthogonal to any diagonal matrix

Question

I need to find a symmetric matrix of real values (not the zero matrix) of any order that is orthogonal to any diagonal matrix of real values. Any hints?

Answer 1

Put zeros on the diagonal and ones on the off-diagonal.

Answer 2

Hint: Using the definition of orthogonality of matrices to be $\text { tr}(AB^t)=0$ (see here), all that matters is what happens on the diagonal of the product matrix $AB^t$.

Think zeros on the diagonal... since $B$ is diagonal, this will result in $AB^t$ having zeros on the diagonal...