defining a quadratic form

Question

Given a diagonal matrix with non-negative entries $D \in \mathbb{R}^{n \times n}$, vector $m_i \in \mathbb{R}^{n}$, $ l_i \geq 0$; I would like to express the following

$$ D \big( \sum_{i=1}^N m_i m_i^T l_i \big) D $$

such that $N < n$ in the following quadratic form

$$ U \Xi U^T $$

where $U \in \mathbb{R}^{n \times N}$ and $\Xi \succeq 0 \in \mathbb{R}^{N \times N}$. I would like to find the pair $(U, \Xi)$ that satisfies this.

Answer 1

Note that $$\sum_{i=1}^N l_i m_i m_i^\top = \sum_{i=1}^N (\sqrt{l_i}m_i)(\sqrt{l_i} m_i^\top) = \begin{bmatrix}\sqrt{l_1}m_1 & \sqrt{l_2}m_2 & \cdots & \sqrt{l_N}m_N\end{bmatrix}\begin{bmatrix}\sqrt{l_1}m_1 \\ \sqrt{l_2}m_2 \\ \vdots \\ \sqrt{l_N}m_N\end{bmatrix}.$$ Since $$\begin{bmatrix}\sqrt{l_1}m_1 \\ \sqrt{l_2}m_2 \\ \vdots \\ \sqrt{l_N}m_N\end{bmatrix}\in\mathbb{R}^{N\times n},$$ you can satisfy your conditions by taking \begin{align*} \Xi &= I_N\succeq 0, \\ U^\top &= \begin{bmatrix}\sqrt{l_1}m_1 \\ \sqrt{l_2}m_2 \\ \vdots \\ \sqrt{l_N}m_N\end{bmatrix} D \in\mathbb{R}^{N\times n}. \end{align*}