# How to perform this vector/matrix computation?

Edit: It seems to me that it could indeed be an outer product. I will keep investigating and update after I reach solid conclusions.

Background: I am trying to implement the algorithm explained here and which consists in, as the title says, A Two-step Kalman/Complementary Filter for Estimation of Vertical Position Using an IMU-Barometer System.

At some point in the paper -after equation 3- the author states that (although the paper is in Korean Google Translator does a good job):

a tilde (~) denotes an outer matrix of the corresponding vector, that is, $\tilde{a}=[a \times]$

If I am not mistaken this means to calculate the cross product of $a$ and the vector following $a$. As an example, assuming both $a$ and $b$ are vectors, then $\tilde{a}b$ would be $a\times b$. (This interpretation is consistent with equation 3, which updates the representation of the earth frame Z-axis in the sensor coordinate system with the angular velocity measurement of the gyro.)

Question: So, if a tilde does in fact mean to calculate the cross product ($\tilde{a}=[a \times]$) how should I perform the following calculation (a simplified version of equation 6.1 in the paper)?

$$Q=\tilde{Z}\Sigma_G \tilde{Z}$$

where both $Z$ and $\Sigma_G$ are $3\times1$ vectors.

Usually such tilde notation for a given vector $z=\begin{bmatrix}z_1&z_2&z_3\end{bmatrix}^\top$ means
$$\tilde{z} = \begin{bmatrix} 0 & -z_3 & z_2 \\ z_3 & 0 & -z_1 \\ -z_2 & z_1 & 0 \end{bmatrix}$$
however if $\Sigma_G \in \mathbb{R}^{3\times1}$, then the product $\Sigma_G\,\tilde{z}$ has a dimension mismatch (the product $\tilde{z}\,\Sigma_G$ is well defined but is also in $\mathbb{R}^{3\times1}$). But you can also interpret the product as
$$\tilde{z}\,\Sigma_G\,\tilde{z} = z \times \Sigma_G \times z$$