Generalized Inverses: Quaternions

A quaternion algebra ℍ $$\mathbb{H}$$ was discovered by Sir Rowan Hamilton in 1843, which is a four-dimensional non-commutative algebra over real number field ℝ $$\mathbb{R}$$ with canonical basis {1, i, j, k} satisfying the conditions: i 2 = j 2 = k 2 = ijk = − 1 , $$\displaystyle{\mathbf{i}^{2} = \mathbf{j}^{2} = \mathbf{k}^{2} = \mathbf{i}\mathbf{j}\mathbf{k} = -1,}$$ so that one has ij = − ji = k , jk = − kj = i , and ki = − ik = j . $$\displaystyle{\mathbf{i}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k},\ \mathbf{j}\mathbf{k} = -\mathbf{k}\mathbf{j} = \mathbf{i},\ \mbox{ and }\ \mathbf{k}\mathbf{i} = -\mathbf{i}\mathbf{k} = \mathbf{j}.}$$ Any element α ∈ ℍ $$\alpha \in \mathbb{H}$$ can be written in a unique way: α = a + b i + c j + d k, where a, b, c, and d are real numbers, i.e., ℍ = { a + b i + c j + d k | a , b , c , d ∈ ℝ } $$\mathbb{H} =\{ a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}\ \vert \ a,b,c,d \in \mathbb{R}\}$$ . The conjugate of α is defined as α ̄ = a − b i − c j − d k $$\bar{\alpha }= a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k}$$ , and the norm | α | is given by | α | = α α ̄ . $$\vert \alpha \vert = \sqrt{\alpha \bar{\alpha }}.$$ It is well-known that ℍ $$\mathbb{H}$$ is a skew field (or called a division ring).