# Question in do Carmo

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?

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At a point, we can view Ric(U,V) as a symmetric bilinear form on the vector space $T_pM$. But symmetric bilinear forms are known to be orthogonally diagonalizable.