Chapter 51: Quaternion Matrices¶
Prerequisites: Clifford Algebra (Ch50) · Matrix Algebra (Ch02) · Eigenvalues (Ch06)
Chapter Outline: From Complex Numbers to Quaternions → Non-commutativity of the Quaternion Algebra \(\mathbb{H}\) → Definitions and Operational Laws of Quaternion Matrices → The Divergence between Left and Right Eigenvalues → The Core Challenge: Non-commutative Determinants (Dieudonné Determinant) → Complex Representation of Quaternion Matrices → Spectral Structure Theorem → Unitary Quaternion Matrices (\(Sp(n)\)) → Applications: Robotic Pose Control, Quaternionic Quantum Mechanics, and Color Image Representation in Signal Processing
Extension: Quaternion matrices are an attempt to place linear algebra on a "non-commutative field"; by introducing three imaginary units \(i, j, k\), they perfectly encapsulate 3D rotation information—the ultimate mathematical framework for processing data with strong rotational correlation.
When we replace the entries of a matrix with Quaternions, we enter a non-commutative algebraic world. Since \(ij \neq ji\), traditional determinant definitions and eigenvalue theories undergo drastic changes. Quaternion Matrices are not only theoretically challenging in pure mathematics but also demonstrate incredible elegance in describing robotic joint movements, spacecraft attitude, and color image processing. This chapter introduces how to re-establish order in this non-commutative domain.
51.1 Foundations of Quaternion Algebra \(\mathbb{H}\)¶
Definition 51.1 (Quaternion)
A quaternion \(q \in \mathbb{H}\) is of the form \(q = a + bi + cj + dk\), satisfying Hamilton's rules: $\(i^2 = j^2 = k^2 = ijk = -1\)$ Key Conflict: \(ij = k\), but \(ji = -k\). Multiplication is not commutative.
51.2 Eigenvalues: Left vs. Right¶
Definition 51.2 (Left and Right Eigenvalues)
For a square quaternion matrix \(A\): 1. Left Eigenvalue: \(A\mathbf{x} = \lambda \mathbf{x}\). 2. Right Eigenvalue: \(A\mathbf{x} = \mathbf{x} \lambda\). Difference: Due to non-commutativity, right eigenvalues are far more research-worthy and exhibit "spectral" properties similar to complex eigenvalues.
51.3 Complex Representation¶
Technique: Mapping to Complex Matrices
Every \(n \times n\) quaternion matrix \(A\) can be mapped to a \(2n \times 2n\) complex matrix \(\mathcal{X}(A)\): $\(q = \alpha + j\beta \to \begin{pmatrix} \alpha & \beta \\ -\bar{\beta} & \bar{\alpha} \end{pmatrix}\)$ This allows us to leverage classical complex linear algebra tools to solve quaternionic problems.
Exercises¶
1. [Basics] Calculate the quaternion product \((1+i)j\).
Solution
Steps: 1. Apply distributivity: \(1 \cdot j + i \cdot j\). 2. Apply Hamilton's rules: \(ij = k\). Conclusion: The result is \(j + k\). Note: \(j(1+i) = j + ji = j - k\), which is different.
2. [Complex Rep] Represent \(q = i\) as a \(2 \times 2\) complex matrix.
Solution
Construction: 1. Write as \(q = \alpha + j\beta\): here \(\alpha = i, \beta = 0\). 2. Apply the formula: \(\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}\). Verification: The square of this matrix is \(-I\), consistent with \(i^2 = -1\).
3. [Determinant] Why can't the Leibniz formula be used directly for quaternion determinants?
Solution
Reasoning: The Leibniz formula involves products of elements (e.g., \(a_{1,\sigma(1)} a_{2,\sigma(2)} \cdots\)). Since quaternion multiplication is non-commutative, changing the order of terms changes the value of the product. Thus, the traditional determinant loses its uniqueness. Mathematicians introduced the Dieudonné determinant to solve this.
4. [Right Eigenvalues] Prove: If \(\lambda\) is a right eigenvalue of \(A\), then \(q^{-1}\lambda q\) is also an eigenvalue for any \(q \in \mathbb{H}, q \neq 0\).
Solution
Proof: 1. Given \(A\mathbf{x} = \mathbf{x}\lambda\). 2. Consider vector \(\mathbf{y} = \mathbf{x}q\). 3. \(A\mathbf{y} = A(\mathbf{x}q) = (A\mathbf{x})q = (\mathbf{x}\lambda)q = \mathbf{x}q(q^{-1}\lambda q) = \mathbf{y}(q^{-1}\lambda q)\). Conclusion: Eigenvalues form "conjugacy classes." This shows that every non-real eigenvalue of a quaternion matrix actually corresponds to a sphere in the 3D imaginary space.
5. [Transpose] Define the conjugate transpose \(A^*\) for quaternion matrices.
Solution
Definition: \((A^*)_{ij} = \overline{a_{ji}}\). Where the quaternion conjugate \(\bar{q} = a - bi - cj - dk\). Note: \(\overline{pq} = \bar{q}\bar{p}\).
6. [Unitary] What is a unitary quaternion matrix (Symplectic matrix)?
Solution
Definition: A quaternion matrix satisfying \(A^* A = I\). These matrices form the Symplectic group \(Sp(n)\), describing symmetries that preserve a specific skew-symmetric structure in physics.
7. [Calculation] Find the conjugate transpose of \(\begin{pmatrix} i & j \\ k & 1 \end{pmatrix}\).
Solution
Steps: 1. Transpose and conjugate: \(\begin{pmatrix} \bar{i} & \bar{k} \\ \bar{j} & \bar{1} \end{pmatrix}\). 2. Substitute conjugates: \(\begin{pmatrix} -i & -k \\ -j & 1 \end{pmatrix}\).
8. [Real Spectra] Prove: Right eigenvalues of a Hermitian quaternion matrix (\(A^*=A\)) must be real.
Solution
Proof Sketch: Use the complex representation \(\mathcal{X}(A)\). If \(A\) is Hermitian, \(\mathcal{X}(A)\) is a complex Hermitian matrix. Since complex Hermitian matrices have real eigenvalues, and quaternion right eigenvalues are contained within the spectrum of \(\mathcal{X}(A)\), the result follows.
9. [Inversion] What is the condition for a quaternion matrix to be invertible?
Solution
Conclusion: Its complex representation \(\mathcal{X}(A)\) must be non-singular. In the quaternion domain, this corresponds to its Dieudonné determinant being non-zero.
10. [Application] Briefly state the advantage of quaternion matrices in color image processing.
Solution
Color images have R, G, B channels. Using pure quaternions \(q = ri + gj + bk\) to represent a pixel treats the image as a quaternion matrix. Advantage: When performing rotations or filtering, quaternion algebra automatically preserves the correlation between R, G, and B components, avoiding color shifts that occur when processing channels independently.
Chapter Summary¶
Quaternion matrices are the pinnacle of non-commutative algebra applications:
- Fusion of Dimensions: Quaternions are not just numbers but geometric entities with internal rotation; quaternion matrices weave 3D rotation perfectly into the framework of linear operators.
- Spherical Spectra: The phenomenon of right eigenvalue conjugacy classes reveals that the spectrum of a non-commutative operator is no longer a set of isolated points but symmetric orbits, greatly enriching the geometry of operator theory.
- Representation Switching: The complex representation technique bridges non-commutative algebra and classical complex analysis, proving that even in environments without commutativity, we can regain control through dimension doubling.