Problem 19 - Complex Analysis

Question

Hamilton's Quaternions (W. R. HAMLLTON, 1843) We consider the following map $$ \begin{array}{r} H: \mathbb{C} \times \mathbb{C} \longrightarrow M(2 \times 2 ; \mathbb{C}) \\ (z, w) \mapsto H(z, w):=\left(\begin{array}{cc} z & -w \\ \bar{w} & \bar{z} \end{array}\right) \end{array} $$ and denote its image by $$ \mathcal{H}:=\\{H(z, w) ; \quad(z, w) \in \mathbb{C} \times \mathbb{C}\\} \subset M(2 \times 2 ; \mathcal{C}) $$ Show that \(\mathcal{H}\) is a skew field, i.e. in \(\mathcal{H}\) all the field axioms hold with the exception of the commutativity law for multiplication. Remark. The notation \(\mathcal{H}\) is intended to remind us of Sir WuLLIAM ROWAN HAMILTON (1805-1865). One calls \(\mathcal{H}\) the HAMILTON quaternions.

Step-by-Step Solution

Answer

The set \( \mathcal{H} \) forms a skew field because it follows field axioms except multiplication commutativity.

Step 1: Verify Closure Under Addition

To verify closure under addition, take two matrices \( H(z_1, w_1) \) and \( H(z_2, w_2) \) from \( \mathcal{H} \). Add these two matrices:\[H(z_1, w_1) + H(z_2, w_2) = \begin{pmatrix} z_1 & -w_1 \ \bar{w}_1 & \bar{z}_1 \end{pmatrix} + \begin{pmatrix} z_2 & -w_2 \ \bar{w}_2 & \bar{z}_2 \end{pmatrix} = \begin{pmatrix} z_1+z_2 & -(w_1+w_2) \ \overline{w_1+w_2} & \overline{z_1+z_2} \end{pmatrix}\]This is of the form \( H(z, w) \) where \( z = z_1 + z_2 \) and \( w = w_1 + w_2 \). Thus, the set \( \mathcal{H} \) is closed under addition.

Step 2: Verify Closure Under Multiplication

Take two matrices \( H(z_1, w_1) \) and \( H(z_2, w_2) \), and multiply:\[H(z_1, w_1) H(z_2, w_2) = \begin{pmatrix} z_1 & -w_1 \ \bar{w}_1 & \bar{z}_1 \end{pmatrix} \begin{pmatrix} z_2 & -w_2 \ \bar{w}_2 & \bar{z}_2 \end{pmatrix} = \begin{pmatrix} z_1z_2 + w_1\bar{w}_2 & -(z_1w_2 + w_1\bar{z}_2) \ \overline{z_1w_2 + w_1\bar{z}_2} & \bar{z}_1\bar{z}_2 + \bar{w}_1w_2 \end{pmatrix}\]The result is another Hamilton quaternion matrix \( H(z, w) \), confirming closure under multiplication.

Step 3: Verify Associativity Under Addition and Multiplication

The sum and product of any three Hamilton quaternion matrices are associative because they are matrices, and addition and multiplication of matrices are associative by their standard algebraic properties.

Step 4: Identify the Existence of Additive Identity

The additive identity in this skew field is \( H(0, 0) = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \), since for any \( H(z, w) \), adding \( H(0, 0) \) yields:\[H(z, w) + H(0, 0) = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix} + \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix}\]

Step 5: Identify the Existence of Additive Inverse

The additive inverse of \( H(z, w) \) is \( H(-z, -w) = \begin{pmatrix} -z & w \ -\bar{w} & -\bar{z} \end{pmatrix} \), such that:\[H(z, w) + H(-z, -w) = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix} + \begin{pmatrix} -z & w \ -\bar{w} & -\bar{z} \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\]

Step 6: Identify the Existence of Multiplicative Identity

The multiplicative identity is \( H(1, 0) = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \), fulfilling:\[H(z, w) H(1, 0) = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix} \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix}\]

Step 7: Identify the Existence of Multiplicative Inverse (except Zero Element)

For any non-zero \( H(z, w) \), the inverse is obtained using:\[H(z, w)^{-1} = \frac{1}{|z|^2 + |w|^2} \begin{pmatrix} \bar{z} & w \ -\bar{w} & z \end{pmatrix}\]This inverse satisfies:\[H(z, w) H(z, w)^{-1} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]confirming existence for non-zero matrices.

Step 8: Confirm Non-Commutativity

To demonstrate non-commutativity, consider two distinct quaternions \( H(z_1, w_1) \) and \( H(z_2, w_2) \), such that:\[H(z_1, w_1) H(z_2, w_2) eq H(z_2, w_2) H(z_1, w_1)\]Generally, unless both matrices are in a special form, their multiplication will not commute, confirming the lack of commutativity.

Key Concepts

Skew Field

A skew field, often referred to as a division ring, is an algebraic structure that is similar to a field but does not require the multiplication operation to be commutative. This means that while every non-zero element has a multiplicative inverse, the order in which you multiply two elements can affect the result. In the context of Hamilton's Quaternions, the set \( \mathcal{H} \) showcases the skew field properties beautifully. It satisfies all of the field axioms, including:

• Closure: Under both addition and multiplication,\( \mathcal{H} \) remains closed, allowing any operations within the set to produce results that also belong to the set.
• Associativity and Identity: It has associative properties and both additive (\( H(0,0) \)) and multiplicative (\( H(1,0) \)) identities are present.
• Existence of Inverses: Every non-zero element has an inverse, ensuring that division is possible.

However, the multiplication in \( \mathcal{H} \) is not commutative, making it a skew field rather than a full field. This property of non-commutativity offers unique advantages in certain mathematical and physical applications, such as rotations in 3D space and theoretical physics.

Matrix Algebra

Matrix algebra involves the manipulation and study of matrices, which are rectangular arrays of numbers arranged in rows and columns. In Hamilton's quaternions, matrix algebra is at the core of transforming complex numbers into a workable form for advanced computations. The quaternions can be represented as matrices like:\[H(z, w) = \begin{pmatrix} z & -w \ \bar{w} & \bar{z} \end{pmatrix}\]This form allows us to explore and confirm essential properties such as closure, identity, and invertibility within matrix algebra.
Matrices provide a powerful way to perform and visualize complex operations, such as rotations and transformations. This is particularly useful in 3D graphics and robotics.

• Operations: Addition and multiplication of matrices follow specific rules that are vital to maintaining algebraic structure and ensuring predictable results.
• Associativity: An important property of matrices where addition and multiplication are consistently associative, ensuring reliable computation when dealing with sequences of matrices.
• Determinants and Inverses: A square matrix in matrix algebra can have an inverse if its determinant is non-zero, which is essential for solving matrix equations and transformations.

Non-Commutative Algebra

Non-commutative algebra refers to any algebraic structure where the commutative property of multiplication (i.e., \( ab = ba \)) does not necessarily hold. Hamilton's quaternions are a prime example of non-commutative algebra. Specifically, when multiplying two quaternion matrices \( H(z_1, w_1) \) and \( H(z_2, w_2) \), the result is different if we change the order of multiplication:\[H(z_1, w_1)H(z_2, w_2) eq H(z_2, w_2)H(z_1, w_1)\]This property leads to intriguing and necessary applications in various domains.

• Complexity: The lack of commutativity introduces complexity which can be leveraged in physics and computer science for tasks like modeling waves, quantum mechanics, and more.
• Visual Representation: In 3D rotations, non-commutativity means the sequence of rotations alters the final orientation, a fundamental concept in computer graphics.
• Broader Applications: Other areas such as cryptography and theoretical mathematics benefit from these properties by utilizing the deeper mathematical structures and behaviors intrinsic to non-commutative operations.