Is there a name for this 'multiplication table' operation of vectors?

Question

Is there a name for an operation or a transformation that generates a matrix from two vectors, creating a multiplication table from its elements? This might be like multiplying a vector $\mathbf{s}$ to another column vector $\mathbf{v}$ as if $\mathbf{s}$ were a scalar.

$$M\left( \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}, \begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix} \right) = \begin{bmatrix} \mathbf{s}v_1 \\ \mathbf{s}v_2 \\ \mathbf{s}v_3 \end{bmatrix} = \begin{bmatrix} s_1v_1 & s_2v_1& s_3v_1 \\ s_1v_2 & s_2v_2 & s_3v_2 \\ s_1v_3 & s_2v_3 & s_3v_3\end {bmatrix}$$

"Multiplication table matrix" and "vector as a scalar" gave me nothing useful in google. If there is no such name I would also appreciate suggestions on a better way to notate this besides writing out the entire matrix.

Answer 1

Let $\mathbf{v} =\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$ and $\mathbf{s} =\begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix}$. Define $$M(\mathbf{v}, \mathbf{s}) =\begin{bmatrix} v_1\mathbf{s} \\ v_2\mathbf{s} \\ v_3\mathbf{s} \end{bmatrix} = \begin{bmatrix} v_1s_1 &v_1s_2& v_1s_3 \\ v_2s_1 & v_2s_2 & v_2s_3 \\ v_3s_1 & v_3s_2 & v_3s_3\end{bmatrix}.$$ Your matrix $M(\mathbf{v}, \mathbf{s})$ happens to be algebraically equal to these expressions:

• the outer product $\mathbf{v} \mathbf{s}^{\rm T}$,
• the unnamed product $\mathbf{v} \operatorname{diag}(\mathbf{s})$ where $\operatorname{diag}(\mathbf{s})$ is a diagonal matrix with the entries of $\mathbf{s}$ along the diagonal, and
• a Kronecker matrix product of the form $\mathbf{v} \otimes \mathbf{s}^{\rm T}$.