$$
\begin{align*}
\sum_{m=1}^r \lambda_m x_m x_m^T + \sum_{m=r+1}^n \lambda_m x_m x_m^T = \sum_{m=1}^r \lambda_m x_m x_m^T
\end{align*}
$$

$$
\begin{align*}
\lambda_{r+1}=\cdots=\lambda_n=0
\end{align*}
$$