decomposing multiplication of two matrices to the sum of rank-1 matrices

Question

Suppose we have two matrices:

$D_{n \times k}$ and $X_{k \times p}$

I need to understand how do we decompose the multiplication DX to the sum of $k$ (am I correct about $k$?) rank_$1$ matrices.

Answer 1

There's no need for the SVD. The point of this question is just to understand the "columns times rows" way of viewing matrix multiplication.

Let $D = \begin{bmatrix} a_1 & a_2 & \cdots & a_k \end{bmatrix}$ and $X = \begin{bmatrix} b_1^T \\ b_2^T \\ \vdots \\ b_k^T \end{bmatrix}$. Then \begin{equation} D X = a_1 b_1^T + \cdots + a_k b_k^T \end{equation} and each term in the sum is a rank $1$ matrix.