# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 3 Notation and Terminology for Geometric Algebra
A \(k\)-blade is any multivector that is expressible as the outer product of \(k\) linearly independent 1-vectors:  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]  
In \(\mathbb{G}^{3}\), all non-zero \(k\)-vectors are \(k\)-blades. Let \(\mathbf{B}=\mathbf{u}_{1}\mathbf{u}_{2}\cdots\mathbf{u}_{k}\) be a geometric product of orthogonal vectors. Then, the reverse of \(\mathbf{B}\) is given as  
\[\mathbf{B}^{\dagger}=\mathbf{u}_{k}\mathbf{u}_{k-1}\cdots\mathbf{u}_{1}. \tag{19}\]  
Suppose we have a vector \(\mathbf{a}\) and a \(k\)-blade \(\mathbf{B}\). Then, the geometric product may be expressed as  
\[\mathbf{a}\mathbf{B}=\mathbf{a}\cdot\mathbf{B}+\mathbf{a}\wedge\mathbf{B} \tag{20}\]  
where the inner (left contraction) and outer products are  
\[\mathbf{a}\cdot\mathbf{B}=\langle\mathbf{a}\mathbf{B}\rangle_{k-1}\;\;,\; \mathbf{a}\wedge\mathbf{B}=\langle\mathbf{a}\mathbf{B}\rangle_{1+k}. \tag{21}\]  
For additional details of GA theory, the reader is referred to the works of [PERSON] [29; 30; 31], [PERSON] [32; 33], [PERSON] et al. [34], and [PERSON] [35]. **Definition 10** (**Rotor**). : _A rotor is a unit norm even grade multivector used to represent a rotation. A rotor \(R\) is any element of \(\mathbb{G}^{n}\) that can be written as the product of an even number of unit vectors and satisfies_  
\[R^{\dagger}R=1. \tag{22}\]  
**Theorem 5** (**Rotation of a Vector**). : _Let \(\mathbf{u}\) be a vector and \(R\) be a rotor; then, the rotation of vector \(\mathbf{u}\) by \(R\) resulting in the vector \(\mathbf{u}^{\prime}\) is expressed as_  
\[\mathbf{u}^{\prime}=R^{-\frac{1}{2}}\mathbf{u}R^{\frac{1}{2}}. \tag{23}\]  
**Definition 11** (**Simple Bivector**). : _Let \(\mathbf{u}\) and \(\mathbf{v}\) be two orthogonal vectors; then, a simple bivector is defined as_  
\[\mathbf{u}\wedge\mathbf{v}=\frac{1}{2}(\mathbf{u}\mathbf{v}-\mathbf{v} \mathbf{u})=\mathbf{u}\mathbf{v}. \tag{24}\]  
_In Euclidean spaces, the square of a simple unit bivector \(\mathbf{j}\) is \(\mathbf{j}^{2}=-1\)._  
**Definition 12** (**Simple Rotor**). : _A rotor, \(R\), that can be expressed as the product of two unit vectors, \(\mathbf{u},\mathbf{v}\), is called a simple rotor [33]._  
\[R=\mathbf{u}\mathbf{v}. \tag{25}\]  
_Any simple rotor \(R\) can be expressed in terms of the exponential of a simple bivector \(\mathbf{B}\)_  
\[R=\mathbf{e}^{\mathbf{B}}=\mathbf{e}^{\mathbf{j}\theta}. \tag{26}\]  
**Corollary 2** (Simple Rotation of a Vector). : _Let \(\mathbf{u}^{\prime}\) denote the result of rotating the vector \(\mathbf{u}\) in the plane \(\mathbf{j}\) by angle \(\theta\). Then, \(\mathbf{u}^{\prime}\) is expressed as_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]0  
Decomposing \(\mathbf{u}\) in the plane \(\mathbf{j}\) in (27) gives  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]1 \[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]2 (28b) \[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]3  
The result in (28c) illustrates how rotation affects the vector \(\mathbf{u}\) within the plane of rotation \(\mathbf{j}\). **Remark 4**. : _Let \(\mathbf{u}^{\prime}\) denote the result of rotating the vector \(\mathbf{u}\) in the plane \(\mathbf{j}\) by angle \(\theta\). Then, the rotation of vector \(\mathbf{u}\) performed from the right side is related to the rotation performed from the left side as_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]4  
**Definition 13** (Polar Representation of a Vector-Valued Signal).