Proof of this theorem: $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1} $

Question

If $A$ is an $n\times n$ symmetric matrix with $r$ nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r$ then

$$ {\rm tr}(A^-)=\sum_{i=1}^r \lambda_i^{-1}. $$

Note: $A^-$ is generalized inverse of $A$.

Answer 1

I assume "generalized inverse" is the Moore-Penrose pseudoinverse. Both sides are invariant under unitary transformations, and they are equal for diagonal matrices.