Is it possible to simplify this expression? Recently I've been stuck in an expression:
$$\sum_{m=1}^{N}{{\lambda}_m {\textbf x_m}^T}\cdot \sum_{m=1}^{N}{{\lambda}_m {\textbf x_m}}$$ where $\textbf x_m\in {\mathbb R}^n$ is a column vector. Could I further simplify this expression to more compact form?

Unless you have some orthogonality conditions on the vectors, not really.
$$
\sum(\ldots) \cdot \sum(\ldots ) =  \sum_{i = 1}^N \sum_{j = 1}^N \lambda_i \lambda_j (x_i \cdot x_j)
$$
Where I've used the dot product instead of multiplying by the transpose. Note that everything commutes, so this is really
$$
\sum_{k = 1}^N \lambda_k^2 \|x_k\|^2 + 2\sum_{i < j}^N \lambda_i \lambda_j (x_i \cdot x_j)
$$
i.e. one copy of every square of things, and two copies of each `non-square' product.
