What kind of multiplication is expected here?

Question

I'm currently trying to implement an algorithm I found in a paper on occupancy detection. There is a certain part which doesn't make sense to me as it seems to have the multiplication order of two vectors reversed. The part is the following;

Let $b_X \neq 0 $ and choose $L_1,L_2$ such that the matrix $A - \begin{pmatrix} L1 \\ L2 \end{pmatrix}C$ is Hurwitz where

$$ \begin{align} &A = \begin{pmatrix} -a & 0 \\ 0 & 0 \end{pmatrix} \\[10pt] &C = (\pi_1(1) \quad \pi_2(1)) \end{align} $$

How would I even multiply $$ \begin{pmatrix} L1 \\ L2 \end{pmatrix} * (\pi_1(1) \quad \pi_2(1))$$

$\pi_1(1)$ and $\pi_2(1)$ both resolve to positive real numbers.

Answer 1

It is clear that all of $L_1,L_2,\pi_1(1),\pi_2(1)$ are scalars. Thus the left multiplicand is a $2×1$ matrix and the right one is a $1×2$ matrix, and standard matrix multiplication produces a $2×2$ matrix, as desired.

The matrix product of a $p×1$ and $1×q$ matrix is known as an outer product. Here $p$ need not equal $q$.

Answer 2

$(2\times 1)\cdot(1\times 2) = 2\times 2$, $$ \binom ab \cdot(c,d) = \begin{pmatrix}ac & ad\\bc & bd\end{pmatrix}. $$