2.1.2. Multi-Vector

Let G*n* = *C<sup>n</sup>*,0, which is the real GA of the quadratic pair (*<sup>V</sup>*, *Q*) where *V* = R*n* and *Q* is the quadratic form of signature (*<sup>n</sup>*, <sup>0</sup>). There is an orthogonal basis {**<sup>e</sup>**1, **e**2,..., **<sup>e</sup>***n*} in R*<sup>n</sup>*, which generates 2*n* basis elements of G*n* via the geometric product as shown in (2):

$$\underbrace{\{1\}}\_{k=0}\underbrace{\{\mathbf{e}\_{i}\}}\_{k=1}\underbrace{\{\mathbf{e}\_{ij},i$$

for *i*, *j* = 1, 2, . . . , *n*.

> The multi-vector *A* of G*n* is defined as

$$\begin{split} A &= E\_0(A) + \sum\_{1 \le i \le n} E\_i(A)\mathbf{e}\_i + \sum\_{1 \le i < j \le n} E\_{ij}(A)\mathbf{e}\_{ij} + \dots + E\_{1\dots n}(A)\mathbf{e}\_{1\dots n} \\ &= \langle A \rangle\_0 + \langle A \rangle\_1 + \langle A \rangle\_2 + \dots + \langle A \rangle\_n \end{split} \tag{3}$$

where *Ei*(*A*), *Eij*(*A*),..., *<sup>E</sup>*1...*n*(*A*) ∈ R, and *Ak* denotes the component of *A* of grade *k*. The reverse of multi-vector *A* is defined as

$$\tilde{A} = \sum\_{k=0}^{n} (-1)^{k(k-1)/2} \langle A \rangle\_k. \tag{4}$$

#### *2.2. The Geometric Algebra of Euclidean 3-Space*

According to the structural characteristics of EMVS, G3 is chosen to model and process the received signals [21]. The multiplication rule can be found in Table 1.

**Table 1.** The multiplication rule in G3.


Referring to (2) and (3), a G3 matrix with *m*-row and *n*-column, noted G*m*×*n* 3 , is constructed as follows [20]

$$\begin{aligned} \mathbf{A} &= \mathbf{A}\_0 + \mathbf{A}\_1 \mathbf{e}\_1 + \mathbf{A}\_2 \mathbf{e}\_2 + \mathbf{A}\_3 \mathbf{e}\_3 + \mathbf{A}\_4 \mathbf{e}\_{12} \\ &+ \mathbf{A}\_5 \mathbf{e}\_{23} + \mathbf{A}\_6 \mathbf{e}\_{13} + \mathbf{A}\_7 \mathbf{e}\_{123} \end{aligned} \tag{5}$$

where **A***k* for *k* = 1, 2, 3, ... , 7 are all *m* × *n* real number matrices. The transpose with reversion of **A** is denoted by **A***<sup>H</sup>*

$$\begin{aligned} \mathbf{A}^H &= \mathbf{A}\_0^T + \mathbf{A}\_1^T \mathbf{e}\_1 + \mathbf{A}\_2^T \mathbf{e}\_2 + \mathbf{A}\_3^T \mathbf{e}\_3 - \mathbf{A}\_4^T \mathbf{e}\_{12} \\ &- \mathbf{A}\_5^T \mathbf{e}\_{23} - \mathbf{A}\_6^T \mathbf{e}\_{13} - \mathbf{A}\_7^T \mathbf{e}\_{123} \end{aligned} \tag{6}$$

where **A***Ti*for *k* = 1, 2, 3, . . . , 7 denotes the transpose.
