In Exercise 9, Chaper 4, do Carmo gives a hint to solving a problem. He says to consider an orthonormal basis $e_1, \ldots, e_n$ in $T_pM$ such that if $x = \sum_{i=1}^n x_i e_i$,
$\text{Ric}_p(x) = \sum \lambda_i x_i^2,$
$\lambda_i$ real. My question is, why must such an orthonormal basis exist?