First off, the question: $A=QLQ'$ given that $Q$ is orthogonal, and $L$ is a real diagonal matrix, is the following true: $A$ is a real symmetric matrix.
My attempts
The easy part is to recognize $Q' = Q^T$. Therefore we have a similarity equation between A and L. This question basically boils down to whether all similar matrices to L is real symmetric. I know that L has all real eigenvalues, and so does A. However, could there exist not-real-symmetric matrices that also have all real eigenvalues?
So first I tried to think of easy contradictory examples but they all came out to be real symmetric. Take this 1 for example:
Take L to be $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$, I know that the characteristic polynomial must be the same, the rank must be the same, and the geometric multiplicities of its eigenvalues(consequently) is also the same.
The charpoly(L) is $(1-\lambda)(-1-\lambda)$, which is $\lambda^2-1$.
Take A to be $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$. The charpoly(A) is $\lambda^2 - 1$ as well. Its rank is also 2.
But argh, this doesn't help! This is a real symmetric!
So I turn to math SO for help - am I missing an obvious example here? If this is actually true, then what's the proof behind it? Thanks!
