If $A$ is a symmetric matrix, show that every eigenvalue of $A$ is nonnegative if and only if $A=B^2$ for some symmetric matrix $B$.
My idea was to make use of the fact that $A$ is symmetric and thus orthogonally diagonalizable (spectral theorem). Therefore, $A$ can be written as $QDQ^T$. However, I am not sure how to continue from there. Can anyone please help me out?
Best Answer
You're on the right track:
Symmetric B can be rewritten as $S \Lambda S^{-1}$ since symmetric matrices are always diagonalizable. So, $B^2 = S \Lambda S^{-1} S \Lambda S^{-1} = S \Lambda^2 S^{-1}$
Therefore, all the eigenvalues are squares of real numbers(property of symmetric matrices), so the eigenvalues are all positive.